Nikita Nikolaev - tspace.library.utoronto.ca · Abstract Abelianisation of Logarithmic Connections Nikita Nikolaev Doctor of Philosophy Graduate Department of Mathematics University

ABELIANISATION OF LOGARITHMIC CONNECTIONS

by

Nikita Nikolaev

A thesis submitted in conformity with the requirementsfor the degree of Doctor of PhilosophyGraduate Department of Mathematics

University of Toronto

c⃝ Copyright 2018 by Nikita Nikolaev

Abstract

Abelianisation of Logarithmic Connections

Nikita Nikolaev

Doctor of Philosophy

Graduate Department of Mathematics

University of Toronto

2018

This thesis studies an equivalence between meromorphic connections of higher rank and abelian

connections. Given a complex curve X and a spectral cover π : Σ → X, we construct a functor

πab : ConnX → ConnΣ, called the abelianisation functor, from some category of connections on X

with logarithmic singularities to some category of abelian connections on Σ, and we prove that πab

is an equivalence of categories. At the level of the corresponding moduli spaces MX,MΣ, which

are known to be holomorphic symplectic varieties, this equivalence recovers a symplectomorphism

constructed by Gaiotto, Moore, Neitzke in their work on Spectral Networks (2013). Moreover,

the moduli space MΣ is a torsor for an algebraic torus, so in fact πab provides a Darboux coordi-

nate system on MX, known as the Fock-Goncharov coordinates constructed in their work on higher

Teichmuller theory (2006).

To prove that πab is an equivalence of categories, we introduce a new concept called the Voros class.

It is a canonical cohomology class in H1 of the base X with values in the nonabelian sheaf Aut(π∗) of

groups of natural automorphisms of the direct image functor π∗. Any 1-cocycle v representing the

Voros class defines a new functor ConnΣ → ConnX by locally deforming the pushforward functor

π∗; the result is an explicit inverse equivalence to πab, called a deabelianisation functor.

We generalise the abelianisation equivalence to the case of quantum connections: these are ℏ-

families of meromorphic connections restricted to a sectorial neighbourhood in ℏ with prescribed

asymptotic regularity. The Schrodinger equation is a quintessential example. The most important

invariant of a quantum connection ∇ is the Higgs field ∇(0) obtained by restricting ∇ to ℏ = 0

(the so-called semiclassical limit). Then abelianisation may be viewed as a natural extension to an

ℏ-family of the spectral line bundle of ∇(0). That is, we show that for a given quantum connection

(E ,∇), the line bundle Eab obtained from E by abelianisation πab restricts at ℏ = 0 to precisely the

ii

spectral line bundle of the Higgs field ∇(0).

Finally, in this thesis we explore the relationship between abelianisation and the WKB method,

which is an asymptotic approximation technique for solving differential equations developed by

physicists in the 1920s and reformulated by Voros in 1983 using the theory of Borel resummation.

We give an algebro-geometric formulation of the WKB method using vector bundle extensions and

splittings. We then show that the output of the WKB analysis is precisely the data used to construct

the abelianisation functor πab.

iii

Declaration of Originality

The research in this thesis was conducted at the Department of Mathematics, University of Toronto.

Section 2.2 contains a construction of a new object, the de Rham-Prym variety; and section 2.4

contains an original result (proposition 2.42) that is central to the main content of this thesis.

Chapters 3, 4, 5 is original material, with the exception of sections 3.1.1 and 3.1.2 which are

review. As described in the Introduction, the main constructions and theorems are in section 3.3,

theorem 3.45 in section 3.5, theorem 4.11 in section 4.4, theorem 4.16 in section 4.5, theorem 5.4

in section 5.1, theorem 5.5 in section 5.2, corollary 5.7 in section 5.3.

iv

Synopsis

1 Introduction 1

2 Meromorphic Connections 8

2.1 Connections and Differential Operators . . . . . . . . . . . . . . . . . . . . . . . . . . 9

2.2 The de Rham-Prym Variety . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

2.3 Nice Connections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

2.4 Linear Algebra of Transverse Flags . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

3 Abelianisation of sl2-Connections 42

3.1 Quadratic Differentials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

3.2 Cech Open Covers of Σ× and X× . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

3.3 The Abelianisation Functor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

3.4 The Deabelianisation Functor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

3.5 Abelianisation Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84

4 Quantum Differential Operators 91

4.1 The Quantum Family . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93

4.2 Quantum Connections and Quantum Differential Operators . . . . . . . . . . . . . . 94

4.3 Quantum Differential Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99

4.4 Quantum Levelt Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107

4.5 The WKB Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112

5 Abelianisation of Quantum sl2-Connections 121

5.1 The Abelianisation Functor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122

5.2 Quantum Abelianisation Equivalence . . . . . . . . . . . . . . . . . . . . . . . . . . . 125

5.3 Abelianisation and the Exact WKB Method . . . . . . . . . . . . . . . . . . . . . . . . 126

Appendix 128

A Spectral Splitting Convergence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129

v

Table of Contents

1 Introduction 1Broad Context . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1Overview of Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2 Meromorphic Connections 82.1 Connections and Differential Operators . . . . . . . . . . . . . . . . . . . . . . . . . . 9

2.1.1 The Category of Meromorphic Connections . . . . . . . . . . . . . . . . . . . 92.2 The de Rham-Prym Variety . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

2.2.1 The Norm Map and the Prym Variety . . . . . . . . . . . . . . . . . . . . . . . 102.2.2 The de Rham-Prym Variety . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

2.3 Nice Connections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162.3.1 Filtered Vector Bundles and Local Systems . . . . . . . . . . . . . . . . . . . . 162.3.2 Levelt Filtration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172.3.3 Local Normal Forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192.3.4 Nice connections. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

2.4 Linear Algebra of Transverse Flags . . . . . . . . . . . . . . . . . . . . . . . . . . . . 262.4.1 Flags and Filtered Transformations . . . . . . . . . . . . . . . . . . . . . . . . 262.4.2 Transverse Flags . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 282.4.3 A Triple of Transverse Flags . . . . . . . . . . . . . . . . . . . . . . . . . . . . 292.4.4 Transverse Flags in Two Dimensions . . . . . . . . . . . . . . . . . . . . . . . 38

3 Abelianisation of sl2-Connections 423.1 Quadratic Differentials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

3.1.1 Quadratic Differentials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 443.1.2 Spectral Curves for Quadratic Differentials . . . . . . . . . . . . . . . . . . . . 473.1.3 The Stokes Foliation F of the Spectral Curve . . . . . . . . . . . . . . . . . . . 51

3.2 Cech Open Covers of Σ× and X× . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 553.2.1 The Levelt Open Cover of Σ× . . . . . . . . . . . . . . . . . . . . . . . . . . . 553.2.2 The Levelt Open Cover of X× . . . . . . . . . . . . . . . . . . . . . . . . . . . 573.2.3 The Stokes Open Cover SΣ . . . . . . . . . . . . . . . . . . . . . . . . . . . . 573.2.4 The Stokes Open Cover of X× . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

3.3 The Abelianisation Functor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 623.3.1 Construction of (Eab,∇ab) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 643.3.2 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

3.4 The Deabelianisation Functor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 703.4.1 The Canonical Cocycle v(∇ab) . . . . . . . . . . . . . . . . . . . . . . . . . . . 713.4.2 Properties of the Canonical Cocycle v(∇ab) . . . . . . . . . . . . . . . . . . . 743.4.3 The Voros Cocycle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77

3.5 Abelianisation Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 843.5.1 Abelianisation with respect to the Stokes Open Cover SX . . . . . . . . . . . . 843.5.2 The Voros Class . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 863.5.3 Abelianisation Theorem. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89

vi

4 Quantum Differential Operators 914.1 The Quantum Family . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 934.2 Quantum Connections and Quantum Differential Operators . . . . . . . . . . . . . . 94

4.2.1 Quantum Connections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 954.2.2 Classical Spectral Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98

4.3 Quantum Differential Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 994.3.1 Local Normal Form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100

4.4 Quantum Levelt Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1074.4.1 Nice Quantum Connections . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111

4.5 The WKB Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1124.5.1 The WKB Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1124.5.2 The Schrodinger Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1154.5.3 Opers and Transverse Extensions . . . . . . . . . . . . . . . . . . . . . . . . . 119

5 Abelianisation of Quantum sl2-Connections 1215.1 The Abelianisation Functor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1225.2 Quantum Abelianisation Equivalence . . . . . . . . . . . . . . . . . . . . . . . . . . . 1255.3 Abelianisation and the Exact WKB Method . . . . . . . . . . . . . . . . . . . . . . . . 126

Appendix 128A Spectral Splitting Convergence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129

A.1 Spectral Splitting | Convergence of the formal solution . . . . . . . . . . . . 129A.2 Quantum Spectral Splitting | Power Series is Gevrey . . . . . . . . . . . . . . 130

vii

chapter 1

Introduction

Broad context

The theory of meromorphic connections is a modern geometric point of view on the local and global

study of systems of ordinary differential equations with meromorphic coefficients. In broadest

terms, this thesis studies at a global level an equivalence between meromorphic connections of

higher rank and meromorphic connections of rank 1. The scope is limited to the simplest possible

type of singularities (logarithmic or regular singularities), and the main results concern one of the

simplest possible geometric setups (vector bundles on curves with structure group SL2(C)).

The primary tool for analysing meromorphic connections in this thesis is to extract certain canoni-

cal invariant filtrations associated with singularities, called Levelt filtrations. Such filtrations were

first introduced by Levelt in his doctoral thesis [Lev61, Theorem 2.2] and subsequently refined

by Simpson in [Sim90, section 3]. They arise due to a basic fundamental fact about singular dif-

ferential equations: solutions near a singular point exhibit various growth behaviour and can be

ordered by growth rate, thus defining a flag in the space of solutions. The basic idea of using such

flag data at the singular points to analyse meromorphic connections was pioneered by Fock and

Goncharov in 2006 [FG06], as well as in essence by the WKB analysis community as early as 1983

with the work of Voros [Vor83]. In 2013, Gaiotto-Moore-Neitzke [GMN13a, GMN13b] came to a

far-reaching realisation that these flag data should be encoded in a flat line bundle on an auxiliary

curve. This is the primary motivation for this thesis.

1

TOC | chapter 1 | 2

Overview of Results

Abelianisation theorem. Roughly speaking, given a smooth complex curve X and a 2-fold cover

π : Σ → X which is adapted to some additionally chosen data on X, we construct a functor

πab : ConnX ConnΣ

called the abelianisation functor. The category ConnX consists of sl2-connections on X with loga-

rithmic singularities and prescribed generic residues, and the category ConnΣ consists of abelian

logarithmic connections on Σ× with accordingly prescribed residues (where Σ× is the curve Σ

punctured at the ramification locus of π). Here and everywhere, by an “abelian connection” we

mean a connection on a line bundle.

For every meromorphic connection in ConnX, we extract its Levelt filtrations. Then the abelianisa-

tion functor πab glues all these filtration data into a single flat line bundle on Σ×. This construction

is presented in detail in section 3.3. The main result is that this functor is in fact an equivalence.

Theorem 1 (Abelianisation Theorem)

There exists a functor πab, called the abelianisation functor, which is an equivalence of categories

πab : ConnX ConnΣ∼ .

Roughly speaking, the abelianisation functor πab takes a “complicated” rank-2 system on X to a

“simpler” abelian system on Σ×. The curve Σ× usually has a more complicated topology than X: in

the simplest non-trivial example, X = P1 and Σ is an elliptic curve. It is precisely this topological

complexity of Σ that allows us to encode all the data of the rank-2 system as the data of holonomies

of the abelian system along the non-trivial cycles of Σ. Crucially, because the connection on Σ is

abelian (i.e., rank-1), this holonomy data is abelian; hence the name “abelianisation”.

Abelianisation theorem | Context. One significance of this theorem may be described as fol-

lows. The moduli spaces of connections MX,MΣ corresponding to the categories ConnX,ConnΣ are

known to be holomorphic symplectic varieties [AB83, Boa99, Boa01]. The abelianisation functor

πab descends to a bijection πab : MX∼−→ MΣ, and this bijection was shown in [GMN13a] to be

a holomorphic symplectomorphism. Moreover, the moduli space MΣ is a torsor for the algebraic

torus (C×)2gΣ , where gΣ is the genus of Σ. As a result, the symplectomorphism πab : MX∼−→ MΣ

TOC | chapter 1 | 3

defines a holomorphic Darboux coordinate system on MX. These coordinates are often called the

Fock-Goncharov coordinates, first described in [FG06].

Another motivation for theorem 1 comes from spectral correspondence, as described in [BNR89,

Hit87] and also [HSW99, chapter 2]. There, Higgs bundles (E , ϕ) on a curve X are put in bijective

correspondence with line bundles on a spectral curve Σ. In the case of Higgs bundles, the analogue

of πab is given by extracting the line eigen-subbundles of ϕ, and the inverse to πab is given simply

by the direct image functor π∗. Theorem 1 is some sense a natural generalisation of the spectral

correspondence to the case of connections.

Abelianisation theorem | the Voros class. Our method of proof of theorem 1 is to construct an

explicit inverse equivalence ConnΣ → ConnX, called a deabelianisation functor. The most important

immediate observation is that, unlike in the case of Higgs bundles, a deabelianisation functor is not

given simply by the direct image functor π∗. Indeed, if (L,∇) is an abelian connection on Σ×,

its pushforward (π∗L, π∗∇) is connection on the curve X× punctured at the branch locus which

has non-trivial monodromy around the branch locus. (In fact, the monodromy of (π∗L, π∗∇) is a

quasi-permutation representation of the cover π : Σ → X, see [Kor04].) As such, (π∗L, π∗∇) does

not admit a holomorphic extension over the branch points. In other words, the image of the direct

image functor π∗ is altogether in the wrong category, so it cannot possibly be an inverse equivalence

to the abelianisation functor πab.

In order to build a deabelianisation functor, we introduce a new concept called Voros class. It is a

cohomology class V ∈ H1(X×,Aut(π∗)

), canonically associated with πab, with values in the non-

abelian sheaf Aut(π∗), which is the sheaf of groups of natural automorphisms of the pushforward

functor π∗ : ConnΣ → Conn×X , where Conn×

X is the category of meromorphic sl2-connections on the

curve X× punctured at the branch locus. Any cocycle v representing V acts on the direct image

functor π∗ by locally deforming it to produce a new functor πv∗ : ConnΣ → Conn×X . A key step in the

proof of theorem 1 is to show that this deformed functor πv∗ actually factors through ConnX:

ConnΣ Conn×X

ConnX

πv∗

πvab

.

The defining feature of the Voros class V is that for any representative v of V , the functor πvab is an

inverse equivalence to πab.

TOC | chapter 1 | 4

Theorem 2

There exists a canonical cohomology class V ∈ H1(X×,Aut(π∗)

)such that for any representative

v ∈ Z1(X×,Aut(π∗)

)of V , the functor πvab is an inverse equivalence to the abelianisation functor πab.

Quantum connections. In chapter 4 and chapter 5, we generalise the abelianisation functor πab

to what we call quantum connections: these are particular families of meromorphic flat connections

depending on a parameter ℏ ∈ C. The most important aspect of the behaviour of this ℏ-family

is the following. At any fixed nonzero value of ℏ, a quantum connection ∇ restricts to a usual

meromorphic connection ∇(ℏ). Whereas at ℏ = 0, the quantum connection ∇ degenerates to a

Higgs field ∇(0), sometimes referred to as the semiclassical limit of ∇.

Quantum connections are similar to the notion of a λ-connection, introduced by Deligne in letters to

Simpson (see, for example, [Sim97, section 4], [Sim94, p.87]), yet different in one crucial respect:

the parameter ℏ is restricted to a sufficiently narrow germ of a sectorial neighbourhood of 0 in A1ℏ

with prescribed asymptotic regularity. For quantum connections, we are able to establish a local

normal form along singularities (proposition 4.10).

Quantum abelianisation theorem. This local normal form result allows us to deduce a funda-

mental fact about quantum connections that is vital to constructing their abelianisation: we deduce

the existence of what we call quantum Levelt filtration. It is a canonical flat filtration of a quantum

connection ∇ associated with a singularity which is uniform in ℏ and with the property that for

each fixed ℏ, it restricts to a usual Levelt filtration of ∇(ℏ). Thus, a quantum Levelt filtration is an

ℏ-family of usual Levelt filtrations.

Just as in the case of usual connections, the abelianisation functor πab for quantum sl2-connections

is constructed by gluing these quantum Levelt filtrations into a flat quantum abelian connection

(the abelianisation line bundle) on a 2-fold cover Σ. One of the main results in this thesis is that the

generalisation of the Abelianisation Theorem to the case of quantum connections.

Theorem 3 (Quantum Abelianisation Theorem)


πab : QConnX QConnΣ∼ .

Here, QConnX,QConnΣ are categories of quantum connections with prescribed residues. An image

TOC | chapter 1 | 5

under the functor πab is called the abelianisation quantum connection.

Extension of spectral data. For quantum connections, the abelianisation quantum connections

bear significance in relation to their semiclassical limit. At ℏ = 0, by means of the spectral corre-

spondene, the Higgs field ∇(0) determines a spectral curve Σ(0) and a spectral line bundle L(0) over

Σ(0). If the double cover Σ is chosen such that Σ∣∣ℏ=0

= Σ(0), then the restriction of the abelianisa-

tion line bundle to ℏ = 0 is canonically isomorphic to the spectral line bundle. In other words, if Σ

extends the classical spectral curve Σ(0) in an appropriate sense, then the abelianisation line bundle

on Σ extends the spectral line bundle.

Theorem 4

The restriction of the abelianisation quantum connection to the punctured classical spectral curve

(Σ(0))× of the Higgs bundle ∇(0) is canonically isomorphic to (L(0), η(0)) where L(0) is the spectral line

bundle of ∇(0), and η(0) is the canonical 1-form on Σ(0).

The WKB method. The most quintessential example of a quantum connection is the (1-dimensional,

stationary) Schrodinger equation. In 1926, a method of asymptotic approximation of solutions

to the Schrodinger equation, called WKB approximation, was developed by physicists Wentzel,

Kramers, and Brillouin (and even earlier and in a more general context by other physicists and

mathematicians; see [Hea62]). Beginning with the work of Voros [Vor83], and later (to mention

only a few) other mathematicians such as Silverstone [Sil85]; Aoki, Kawai, Takei [AKT91, KT05];

as well as Delabaere, Dillinger, Pham [DDP93], the WKB approximation was reformulated based

on the theory of Borel resummation [Cos09]. In this form, it is known as the exact WKB analysis.

In this thesis, we give an algebro-geometric formulation of the WKB method. Given a flat vector

bundle E which is an extension,

0 E ′ E E ′′ 0 ,

we interpret the WKB problem as a search for an invariant splitting of this sequence. Upon a

choice of a reference (non-invariant) splitting, this search amounts to finding a unipotent gauge

transformationW of E . Using the unipotent nature ofW , this search reduces the problem to solving

a differential equation of lower order, called Riccati equation. We call such a unipotent splitting W

a WKB splitting.

TOC | chapter 1 | 6

Theorem 5 (the WKB ansatz)

If W : E ′′ → E is a WKB splitting, then W (E ′′) ⊂ E is a ∇-invariant subbundle.

In section 5.3, we describe the relationship between abelianisation and the WKB method. We show

that given a Schrodinger connection (E ,∇) (i.e., an quantum sl2-connection on the filtered vector

bundle E which is the jet bundle J 1ℏ L of a certain line bundle L), the invariant splittings obtained

via the WKB method are nothing but the quantum Levelt data.

Theorem 6

If (E ,∇) is a Schrodinger connection and W is a WKB splitting defined in a neighbourhood of the

singular divisor p, then W (L) is naturally isomorphic to a piece Lpi of the associated graded of the

quantum Levelt filtration Ep• on E associated to p.

Further details of the relationship between the WKB analysis and abelianisation of quantum con-

nections will be provided elsewhere.

TOC | chapter 1 | 7

AcknowledgementsFirst and foremost, my deepest gratitude and debt goes to Marco Gualtieri, my doctoral advisor,teacher, mentor, and, above all, my friend. Marco, your help, support, and guidance stretched farbeyond for what I could ever have asked. Your lessons, criticism, insights, and advice: I will carrythem with me for the rest of my career and impart them to future generations of mathematicians.

I would like to express my gratitude also to the members of my supervisory and PhD exam commit-tees: Lisa Jeffrey, Eckhard Meinrenken, as well as Dror Bar-Natan, John Bland, and Yael Karshon.Your questions and suggestions impelled me to be critical about my work, enriching and contribut-ing to my research experience.

Throughout the years of my doctoral programme, I have benefited immensely from fascinatingconversations with many mathematicians. Amongst many others, I wish to mention Anton Alekseev,Francis Bischoff, Olivia Dumitrescu, Aaron Fenyes, Jonathan Fisher, Laura Fredrickson, Kohei Iwaki,Omar Kidwai, Charlotte Kirchhoff-Lukat, Kevin Luk, Mykola Matviichuk, Victor Mouquin, MotohicoMulase, Andrew Neitzke, Brent Pym, Shinji Sasaki, Geoffrey Scott, Artan Sheshmani, and AlanThompson. Special thanks go out to Alberto Garcıa Raboso and Steven Rayan: I wouldn’t be whereI am today without your continual mentorship, support, and encouragement; you persisted withyour belief in me even when I had none. I want to express special gratitude also to Sergey Arkhipov:you drew me into mathematics and witnessed my growth as a mathematician from early days; Ihold dear your continued friendship and collaboration.

I would also like to thank Motohico Mulase and Andrew Neitzke for their hospitality as well asinspiring and stimulating conversations during my respective visits at the University of Californiaat Davis in 2015 and at the University of Texas at Austin in 2016. Both of these trips were fruitfuland bore significant impact on the work accomplished in this thesis. I wish to acknowledge thesupport from the US National Science Foundation that made these visits a reality.

This thesis was made possible in part due to the generous financial support from the University ofToronto; the Natural Sciences and Engineering Research Council of Canada; as well as the OntarioMinistry of Training, Colleges, and Universities.

I would like to thank several other faculty members in the Department of Mathematics who havemade a notable impact on my views about teaching and academic life in general. I would like tomention James Arthur, Almut Burchard, Alfonso Gracia-Saz, Mary Pugh, and Joe Repka. I wouldalso like to thank Ashley Armogan, Sonja Injac, Diana Leonardo, and Jemima Merisca for theirprofessionalism and hard work in ensuring that the administrative side of our department runs likea clock. Finally, I would be remiss not to mention that Ida Bulat’s kindness and optimism transcendsmy years at the University of Toronto; her presence is greatly missed.

The life as a PhD student can at times be testing. Friendship is where one often seeks refuge, and Ihave certainly been fortunate enough to have several wonderful people to call friends. In additionto those already mentioned, I wish to thank Tracey Balehowsky, Fulgencio Lopez, Mario Moscovici,Nikita Pchelin, Patrick Robinson, Nika Shakiba, Hans Schmidt, and Yvon Verberne.

Finally, I wish to thank my entire family for their love and kindness. To my parents, Larisa andAndrey: thank you for your help and generosity. To my brother, Ivan: thank you for instilling in memy sense of humour. To Ingrid Lameda: thank you for your help and support. Lastly, there are notenough words to describe how thankful I am to Beatriz Navarro Lameda: none of this would havehappened without your unwavering patience, encouragement, support, friendship, understanding,and, above all, love. Beatriz, this thesis is as much yours as it is mine.

chapter 2

Meromorphic Connections

Contents2.1 Connections and Differential Operators . . . . . . . . . . . . . . . . . . . . . . . 9

2.1.1 The Category of Meromorphic Connections . . . . . . . . . . . . . . . . . . 9

2.2 The de Rham-Prym Variety . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

2.2.1 The Norm Map and the Prym Variety . . . . . . . . . . . . . . . . . . . . . . 10

The Norm Map. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

The Prym Variety. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

2.2.2 The de Rham-Prym Variety . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

The Norm Map. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

2.3 Nice Connections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

2.3.1 Filtered Vector Bundles and Local Systems . . . . . . . . . . . . . . . . . . . 16

2.3.2 Levelt Filtration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

Levelt exponents. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

Levelt filtration. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

2.3.3 Local Normal Forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

2.3.4 Nice connections. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

Residue data. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

Connections with fixed residue data. . . . . . . . . . . . . . . . . . . 23

Generic residue data. . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

Existence of Levelt filtration. . . . . . . . . . . . . . . . . . . . . . . . 24

Nice connections with fixed residue data. . . . . . . . . . . . . . . . . 24

Abelianisation of generic residue data. . . . . . . . . . . . . . . . . . 26

2.4 Linear Algebra of Transverse Flags . . . . . . . . . . . . . . . . . . . . . . . . . 26

2.4.1 Flags and Filtered Transformations . . . . . . . . . . . . . . . . . . . . . . . 26

2.4.2 Transverse Flags . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

A pair of transverse flags. . . . . . . . . . . . . . . . . . . . . . . . . 28

2.4.3 A Triple of Transverse Flags . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

Two different identifications of associated graded pieces. . . . . . . . 29

A Triple of Transverse Flags. . . . . . . . . . . . . . . . . . . . . . . . 31

Examples of gluing isomorphisms. . . . . . . . . . . . . . . . . . . . . 32

Gluing isomorphisms in general. . . . . . . . . . . . . . . . . . . . . 36

2.4.4 Transverse Flags in Two Dimensions . . . . . . . . . . . . . . . . . . . . . . 38

A pair of transverse flags. . . . . . . . . . . . . . . . . . . . . . . . . 38

A triple of transverse flags. . . . . . . . . . . . . . . . . . . . . . . . . 39

8

TOC | chapter 2 | section 1 9

2.1. Connections and Differential Operators

We being by recalling the definition of a meromorphic connection. References for this standard

material include [Boa99, Sab, Zol06]; see also [Pym13, GLP15] where connections are viewed

from the point of view of representations of Lie algebroids on curves.

2.1.1. The Category of Meromorphic Connections

Let X be a smooth complex projective curve, and D an effective divisor on X.

Definition 2.1 (meromorphic connection)

A connection on (X,D) is an OX-module E together with a CX-linear morphism

∇ : E −→ E ⊗OX

Ω1X(D) .

satisfying the Leibniz rule:

∇(fe) = df ⊗ e+ f∇e .

In other words, (E ,∇) is meromorphic connection on X with poles bounded by D. Here, d is the

usual de Rham differential, but viewed as map

d : OX −→ Ω1X(D)

via the inclusion Ω1X → Ω1

X(D). A morphism φ : (E ,∇) → (E ′,∇′) of two connections on (X,D) is

an OX-linear map φ : E → E ′ intertwining the connections:

E E ⊗ Ω1X(D)

E ′ E ′ ⊗ Ω1X(D) .

∇

φ φ⊗1

∇′

(2.1)

The category of connections on (X,D) will be denoted by Conn(X,D). The subcategory of connec-

tions on (X,D) on vector bundles of rank n will be denoted by Conn(X,D; gln). The corresponding

moduli spaces (i.e., sets of equivalence classes) will be denoted by MdR(X,D) and MdR(X,D; gln),

respectively.


Definition 2.2 (canonical module [GLP15, example 2.11])

Let X be a complex curve and D an effective divisor X. The canonical module for (X,D) is the

meromorphic connection(OX(−D),∇D

), where ∇D is defined as follows. Let p ∈ D be a point

with multiplicity m and x a local coordinate centred at p. The section f(x) = xm ∈ OX(−D) is a

local trivialisation, in which the connection ∇D is given by

∇Df = df = mdx

x⊗ f .

Definition 2.3 (sln-connection)

A connection (E ,∇) on (X,D) of rank n is an sln-connection if the corresponding trace connection

on the determinant bundle (det E , tr∇) is isomorphic to the canonical module(OX(−D),∇D

).

Note that for us the choice of isomorphism (det E , tr∇) ∼−→(OX(−D),∇D

)is not part of the data

of an sln-connection. However, once such a choice µ : det E ∼−→ OX(−D) is made, it can be thought

of equivalently as a nowhere vanishing ∇-invariant global section µ ∈ H0X(det E(D)), also known as

a volume form. The subcategory of sln-connections on (X,D) will be denoted by Conn(X,D; sln).

2.2. The de Rham-Prym Variety

2.2.1. The Norm Map and the Prym Variety

In this section, we recall the classical notion of the norm map and the Prym variety associated with

a finite cover π : Σ → X of algebraic curves. This account closely follows [HP12, section 3] and

[Gro61, section 6.5.1].

1. The Norm Map. Let X be a smooth projective curve, and let π : Σ → X be a finite cover of

degree n. The pushforward π∗OΣ of the structure sheaf is naturally endowed with the structure of

an OX-algebra. Thus, for any local section s ∈ π∗OΣ over U ⊂ X, multiplication by s is an OX-linear

morphism ms : π∗OΣ|U → π∗OΣ|U. As such, it has a well-defined determinant det(ms) ∈ OU.

Definition 2.4 (norm of a section)

The norm of a local section s ∈ π∗OΣ is defined by

Nπ(s) := det(ms) .


The following simple properties of the norm map are easily verifiable.

Lemma 2.5

The norm satisfies the following properties:

Nπ(s1s2) = Nπ(s1)Nπ(s2) for any s1, s2 ∈ π∗OΣ;

Nπ(fs) = fnNπ(s) for any s ∈ π∗OΣ, f ∈ OX;

s ∈ π∗O×Σ ⇔ Nπ(s) ∈ O×

X for any s ∈ π∗OΣ.

2. The Prym Variety. Let L be an invertible OΣ-module. We view the pushforward π∗L as an

invertible π∗OΣ-module on X, and we locally trivialise it accordance with this structure; that is, we

choose an open cover Uα of X together with isomorphisms

φα : π∗L∣∣Uα

∼−→ π∗OΣ∣∣Uα

.

Remark 2.6

Notice that these are not local trivialisations of π∗L in the usual sense as an OX-module. For

example, if L is a line bundle (i.e., locally free of rank 1), then the pushforward π∗L as an OX-

module is locally free of rank n, but it has rank 1 as a π∗OΣ-module (at least away from the branch

locus of π).

The transition data φαβ := φα φ−1β is a local section of π∗O×

Σ over Uαβ, so it has a well-defined

norm Nπ(φαβ) ∈ O×Uαβ

. Since the transition data φαβ form a Cech 1-cocycle φαβ ∈ H1(X, π∗O×Σ ),

their norms form a Cech 1-cocycle Nπ(φαβ) ∈ H1(X,O×X ), determining an invertible sheafNπ(L)

on X, which we call the norm of L.

Lemma 2.7

The norm satisfies the following properties:

Nπ(L1 ⊗ L2) = Nπ(L1)⊗Nπ(L2) and Nπ(π∗M) = M⊗n ,

for any invertible sheaves L1,L2 on Σ, and any invertible sheaf M on X.


In particular, lemma 2.7 says that the above construction produces a natural group homomorphism,

Nπ : Pic(Σ) −→ Pic(X)

L 7−→ Nπ(L) ,

called the norm homomorphism. It restricts to a map Pic0(Σ) −→ Pic0(X)

Definition 2.8 (Prym variety)

The Prym variety associated to π : Σ → X is defined as the subgroup of the identity connected

component Pic0(Σ) of the Picard:

Prym(π) := ker(Nπ : Pic0(Σ) −→ Pic0(X)

).

2.2.2. The de Rham-Prym Variety

In this section, we generalise the notion of the norm of a section to connections. This allows us

to define the analogue of the Prym variety inside the de Rham moduli space of meromorphic flat

connections, which we call the de Rham-Prym variety.

1. The Norm Map. Let X be a smooth projective curve, and let π : Σ → X be a finite cover of

degree n. Let R := Ram(π) ⊂ Σ be the ramification locus of π. Let C,D be effective divisors on Σ,X

satisfying the following relationship:

R ⩽ π∗D and C = π∗D− R . (2.2)

This implies that Ω1Σ(C) = π∗Ω1

X(D).

Remark 2.9 (etale algebroid morphisms)

The twisted tangent sheaves TΣ(−C), TX(−D) have the structure of Lie algebroids: the anchor


maps are the inclusion morphisms TΣ(−C) → TΣ, TX(−D) → TX. The pushforward π∗ : TΣ(−C) →

TX(−D) is a Lie algebroid morphism, and the condition (2.2) on the divisors implies that π∗ is etale

in the terminology of [GLP15, Definition 2.4].

Consider a local section A ∈ Ω1Σ(C) over an open set of the form π−1(U) for some open U ⊂ X. It

can be viewed as a twisted endomorphism mA : OΣ → OΣ⊗Ω1Σ(C) = Ω1

Σ(C) over π−1(U), which is

just multiplication by A. Then the direct image MA := π∗mA is an OU-linear morphism π∗OΣ −→

π∗(Ω1Σ(C) ⊗ OΣ). Using the projection formula together with the fact that Ω1

Σ(C) = π∗Ω1X(D), we

conclude that MA is a twisted endomorphism π∗OΣ −→ Ω1X(D)⊗ π∗OΣ over U ⊂ X. As such, it has

a well-defined trace tr(MA) ∈ Ω1X(D).

Definition 2.10 (norm of a differential form)

Suppose π : Σ → X is such that TΣ(−C) → TX(−D) is an etale Lie algebroid morphism, and

consider the direct image sheaf π∗(Ω1Σ(C)

). The norm of a local section A ∈ π∗

(Ω1Σ(C)

)is the local

section of Ω1X(D) defined by

Nπ(A) := tr(MA) .

Example 2.11

Let Σ := A1z, X := A1

x, and π : Σ → X be the map z 7→ x = z2. It lifts to an etale morphism

TΣ(−1 · 0) → TX(−1 · 0). Consider the 1-form A := a(z) dz /z ∈ Ω1Σ(1 · 0). To compute MA = π∗mA

in the OX-basis (1, z) for π∗OΣ, we first decompose a = a0 + za1 with a0, a1 ∈ OX, and then write

mA(1) = a(z)dz

z= (a0 + za1)

1

2

dx

x,

mA(z) = a(z)zdz

z= (xa1 + za0)

1

2

dx

x.

Therefore,

MA =1

2

a0 xa1

a1 a0

dxx

,

and hence the norm of A is Nπ(A) = a0 dx /x.

Consider now an abelian flat meromorphic connection (L,∇) on (Σ,C). Choose an open cover

Uα of X along with trivialisations

φα : π∗L∣∣Uα

∼−→ π∗OΣ∣∣Uα

.


We have

π∗∇ : π∗L −→ π∗L ⊗ Ω1X(D) .

Then

φα(π∗∇)φ−1α = d +Aα

for some local section Aα of Ω1X(D) ⊗ End(π∗OΣ) over Uα. Define a meromorphic connection on

the structure sheaf OUα by

Nπ∇∣∣Uα

:= d + trAα .

Now, consider another trivialising open neighbourhood (Uβ, φβ). Let Aβ be the corresponding local

section of Ω1X(D)⊗ End(π∗OΣ) over Uβ. Then on the double overlap Uαβ := Uα ∩ Uβ, we have

Aβ = φ−1βαAαφβα + dlogφβα .

Using the properties of trace, we find

trAβ = tr(φ−1βαAαφβα

)+ tr

(dlogφβα

)= trAα + dlog detφβα .

Thus, we find that the two locally defined meromorphic connections Nπ∇|Uαand Nπ∇|Uβ

glue to

a meromorphic connection Nπ∇ on the norm Nπ(L) over Uα ∪ Uβ:

Nπ∇∣∣Uα

= Nπ∇∣∣Uβ

+ dlogNπ(φαβ) .

Therefore, given a finite map π : Σ → X of degree n such that the induced Lie algebroid morphism

TΣ(−C) → TX(−D) is etale, we obtain the de Rham norm map

M1dR(Σ,C) M1

dR(X,D)

(L,∇)(Nπ(L), Nπ(∇)

),

Nπ

where M1dR(Σ,C) is the de Rham moduli space of rank-1 meromorphic connections on Σ with poles

bounded by C. Abelian connections on (X,D) and (Σ,C) form groups, called Picard groups for the

corresponding Lie algebroids. It is straightforward to check that the de Rham norm map Nπ is a

group homomorphism.


Definition 2.12 (de Rham-Prym variety; odd connection)

We define the de Rham-Prym variety by:

DPrym(π) := ker(Nπ

)⊂ M1

dR(Σ,C) .

A connection (L,∇) is an odd connection if its isomorphism class belongs to DPrym(π).

Lemma 2.13

If π : Σ → X is a 2-fold cover with simple ramification, then any odd connection (L,∇) on Σ has

residue −1/2 at every ramification point. In particular, the monodromy of (L,∇) around every rami-

fication point is −1.

Proof.

Near a ∈ R, the cover π : Σ → X is isomorphic to the ramified double cover π : Dz → Dx sending

z 7→ x = z2. Choose a trivialisation φ : L|Dz

∼−→ ODz such that

φ∇φ−1 = d + adz

z,

where a := Resa∇. Now, consider the pushforward (π∗L, π∗∇).

∇1 =a

zdz = a

dx

2x

∇z = dz + adz = (1 + a) dz = z(1 + a)dx

2x.

Thus,

π∗∇ = d +

a 0

0 1 + a

dx2x

,

and hence

Nπ(∇) = d +(a+ 1

2

)dxx

.

Therefore, the connection Nπ(∇) is equivalent to the trivial connection d if and only if the residue

Res0Nπ(∇) = 0; i.e., if and only if a = −12 .

Given an effective divisor R on Σ, there is a canonical abelian meromorphic flat connection

∇R : OΣ → Ω1Σ(R) ,


for which the canonical section sR : OΣ → OΣ(R) is flat. If R is reduced, then in a local coordinate

(U, z) centred at a point in R, this connection has a simple expression

∇R∣∣U= d− dz

z,

where d is the de Rham differential on OΣ.

2.3. Nice Connections

2.3.1. Filtered Vector Bundles and Local Systems

In this thesis, we will be dealing with filtrations on connections and their corresponding local

systems. Our definitions follow [Sim90].

Definition 2.14 (filtered vector bundle)

A filtered vector bundle on X is a vector bundle E together with a filtration by subbundles

E• :=(0 =: E0 ⊂ E1 ⊂ · · · ⊂ En := E

).

The filtration E• is a full filtration, if E has rank n and the filtration E• has the additional property

that rank(Ei) = i.

Definition 2.15 (filtered local system)

A filtered local system on X is a local system V of rank n together with a filtration

V• :=(0 =: V0 ⊂ V1 ⊂ · · · ⊂ Vn := V

),

with the property that rank(Vi) = i.

Given a point p ∈ X, a local system V is a filtered local system near p if p has a sectorial neigh-

bourhood U such that the restriction V|U is a filtered local system Vp• with the additional property

that the filtration Vp• is invariant under the monodromy of V around p.


Definition 2.16 (filtered connection)

A filtered connection on X is a connection (E ,∇) on a vector bundle E equipped with a ∇-invariant

filtration E•.

Given a point p ∈ X, a connection (E ,∇) is a filtered connection near p if p has a sectorial

neighbourhood U such that the restriction (E ,∇)|U is a filtered connection with the additional

property that the filtration Ep• is invariant under the monodromy of ∇ around p.

This means that the restriction of ∇ to each subbundle Ei is a connection (Ei,∇i).

2.3.2. Levelt Filtration

In this thesis, the most important kind of filtration on a connection is a filtration by relative growth

of solutions at singular points. This kind of filtration was first systematically studied by Levelt

[Lev61], and later refined by Simpson [Sim90]. Our approach in this thesis is similar to that of

Simpson. We define the Levelt filtration by defining the Levelt exponents. See [HS99, Chapter VII],

where Levelt exponents are referred to as Liapounoff ’s type numbers.

Let (E ,∇) be a connection of rank n on (X,D). Let V := ker (∇) be the local system of flat sections

of ∇ on X× := X \ D.

1. Levelt exponents. Choose a regular singular point p ∈ D, and let (U, x) be a local coordinate

chart vanishing at p that does not contain any other point of D. Fix any nearby point p′ ∈ U× :=

U\p, and consider the stalk V := Vp′ . Then V is the n-dimensional vector space of ∇-flat sections

of E over any simply connected neighbourhood of p′ contained in U×.

Definition 2.17 (Levelt exponent)

For any nonzero ∇-flat section ψ ∈ V, we define the Levelt exponent of ψ by

λ1p(ψ) := inf

µ ∈ R

∣∣∣ limx→0

(x−µ+εψ

)= 0 for all ε > 0

.

Denote the totally ordered set of Levelt exponents of ∇ at p by

Λ1p(∇) :=

λ1p(ψ)

∣∣∣ ψ ∈ V nonzero⊂ R ,

with elements ordered in an increasing manner.


The point p is assumed to be a regular singularity of ∇, so by definition any ∇-flat section has

at most polynomial growth in any sector with vertex p. The regular Levelt exponents measure

this growth rate. They are real numbers which are independent of the choice of coordinate x,

trivialisation of E , or the choice of a nearby point p′. Furthermore, because the growth of flat

sections near the regular singular point p is at most polynomial, regular Levelt exponents are also

independent of the angle at which p is approached in the limit defining λ1p. This last property does

not hold at an irregular singular point.

Next, we state some elementary properties of Levelt exponents whose verification is immediate

from the definition.

Lemma 2.18 (regular Levelt exponents | elementary properties)

Denote λ1p(ψ) by λ(ψ).

(1) λ(ψ1 + ψ2) ⩽ maxλ(ψ1), λ(ψ2).

(2) If λ(ψ1) > λ(ψ2), then λ(ψ1 + ψ2) = λ(ψ1).

(3) If ψ1, . . . , ψk ∈ V are linearly independent, then λ(ψ1), . . . , λ(ψk) are all distinct.

(4) The set Λ1p(∇) is nonempty of size at most n.

Notice that the converse to property (3) does not hold, as it would contradict property (2).

2. Levelt filtration. Let Λ1p(∇) = λ1 < . . . < λm with 1 ⩽ m ⩽ n. For each i = 1, . . . ,m, define

a subspace of V consisting of all ∇-flat sections with growth rate at most λi:

Vpi :=

ψ ∈ V

∣∣∣ λ1p(ψ) ⩽ λi

.

By properties (1) and (3) of lemma 2.18, Vi is indeed a vector space. Furthermore, from lemma 2.18

it is easy to see that these spaces V1, . . . ,Vm fit into a (possibly partial) filtration of the vector space

V of ∇-flat sections:

Vp• :=

(0 =: Vp

0 ⊂ Vp1 ⊂ · · · ⊂ Vp

m = V)

.

Lemma 2.19

The filtration Vp• is invariant under the monodromy of ∇ around p.

As V is the stalk at p′ of the local system V = ker(∇), lemma 2.19 therefore implies that the filtration


Vp• equips the local system V with the structure of a filtered local system Vp

• near p. Equivalently, it

equips the meromorphic connection (E ,∇) with the structure of a filtered connection Ep• near p.

Definition 2.20 (Levelt filtration)

Let (E ,∇) be a connection on (X,D). Let p ∈ D be a logarithmic singularity, and Λ1p(∇) the

corresponding set of regular Levelt exponents. The resulting filtration Ep• near p will be called the

Levelt filtration near p.

Definition 2.21 (Levelt basis)

If Ep• is a Levelt filtration, any basis of local generators of E near p that is adapted to the filtration

Ep• is a Levelt basis.

2.3.3. Local Normal Forms

Theorem 2.22 (Local Normal Form [Was76, Theorems 5.1, 5.4] )

Let α be a logarithmic differential system

α = A(x)dx

x=

∞∑k=0

Akxk dx

x

whose residue term A0 satisfies the following properties:

(1) A0 is diagonalisable with distinct eigenvalues;

(2) the eigenvalues of A0 have distinct real parts;

(3) no two eigenvalues of A0 differ by an integer.

In other words, A0 ∈ gln is a regular semisimple element that is generic and non-resonant.

Then there exists a gauge transformation G(x), holomorphic at x = 0 with the same radius of conver-

gence as A(x), which brings the differential system α to the simpler diagonal form

α := Λ

dx

x,

where Λ := diag(λ1, . . . , λn) is the diagonal matrix of eigenvalues of A0, which are ordered by their

real parts as Re(λ1) < · · · < Re(λn).

Wasow’s textbook [Was76] contains a very clear and instructive proof of this fundamental fact


about singular differential equations. Here, we provide only a brief outline of its proof. This is

done for completeness but also because in section 4.3 we will prove an analogous local normal

form result for quantum connections using a similar strategy. Finally, we would like to point out

that local normal form results of this kind for differential systems hold in far greater generality,

but theorem 2.22 is sufficient for our purposes. The interested reader is encouraged to consult the

references for further details; Wasow’s textbook [Was76] is an excellent place to start.

Outline and method of proof of theorem 2.22

The action of any gauge transformation G on the differential system α is

G[α] = G−1αG+G−1dG =

(G−1AG+ xG−1 d

dxG

)dx

x.

Thus, to prove the theorem, we wish to find a transformation G = G(x) which satisfies the differ-

ential equation

xd

dxG = GΛ−AG . (2.3)

The construction of a solution to (2.3) consists of two main steps. First, we construct a formal

solution G; i.e., a solution in the form of a power series with a priori no specified convergence

properties. Second, we show that the constructed power series G is in fact a convergent power

series with the desired radius of convergence.

1. Construction of a formal solution. Let

G(x) :=

∞∑k=0

Gkxk . (2.4)

be an element of the ring GLn(CJxK) of invertible n × n matrices with coefficients in the ring of

the formal power series CJxK. Substituting the power series (2.4) into equation 2.3 and separating


order by order in x, we find:

x0 | 0 = G0Λ−A0G0

i.e., Λ = G−10 A0G0 (2.5)

x1 | G1 = G1Λ−A0G1 −A1G0

i.e., (A0 + 1)G1 −G1Λ = −A1G0

x2 | 2G2 = G2Λ−A0G2 −A1G1 −A2G0

i.e., (A0 + 21)G2 −G2Λ = −A2G0 −A1G1 ,

and more generally

xk | (A0 + k1)Gk −GkΛ = −k−1∑s=0

Ak−sGs . (2.6)

Recall that Λ is the matrix of eigenvalues of A0, and by assumption A0 is diagonalisable with

distinct eigenvalues, so equation (2.5) has a unique solution.

To solve each equation (2.6), we recall the following general fact about from linear algebra.

Lemma 2.23 ( [Was76, Theorem 4.1] )

Let A,B,C ∈ End(Cn), and suppose that A,B have no eigenvalues in common. Then the equation

AX −XB = C

has a unique solution for X.

In our case, X is Gk, A is (A0 + k1), B is Λ, and C is the righthand side of (2.6). By assumption,

no two eigenvalues of A0 differ by an integer, so (A0 + k1) and Λ are guaranteed to have no

eigenvalues in common for every integer k. It follows that there is a unique solution Gk of equation

(2.6) for every k. Therefore, the construction of G is complete.

2. Convergence of the formal solution. To prove that G is in fact a convergent power series,

we view the differential equation (2.3) in the following way. We view G as a column vector ψ

in C2n with coefficients that are formal power series in x. This vector ψ is formal solution to

the differential equation x∂xψ = Fψ for some holomorphic matrix F whose entries are linear

combinations of entries of A. Then the theorem follows from the following general lemma.


Lemma 2.24 ( [Was76, Theorem 5.3] )

Let F = F (x) ∈ End(Cm) ⊗ Cx be a holomorphic matrix germ with radius of convergence r > 0.

Suppose ψ ∈ C2n ⊗ CJxK is a formal solution to the following differential equation for ψ:

xd

dxψ = Fψ .

Then ψ is holomorphic at x = 0 with radius of convergence r.

This completes the proof of theorem 2.22.

2.3.4. Nice connections.

1. Residue data. Let (E ,∇) be a connection on (X,D). Then ∇ has a well-defined restriction

∇∣∣D: E∣∣D−→ E

∣∣D⊗OD

Ω1X(D)

∣∣D

,

which is an OD-linear map. Indeed, if a component D′ ⊂ D is locally given by xr = 0, then

the Leibniz rule for the natural vector field xr∂x gives ∇xr∂x(fe) = (xr∂xf)e + f∇xr∂xe (where

f ∈ OX, e ∈ E); restricting to D′ then kills the first term. So ∇|D defines a section

∇∣∣D∈ End

(E |D)⊗ Ω1

X(D)∣∣D

,

called the polar Higgs field [GLP15, §6]. If D is reduced (so ∇ is a logarithmic connection), then

we have the residue sequence:

0 Ω1X Ω1

X(D)m⊕i=1

C 0 ,Res (2.7)

where m = |D| is the number of points in D. So for every point p ∈ D, we have

Ω1X(D)

∣∣p

∼−→ C ,

where the isomorphism is given by the residue map in (2.7). Moreover, we can choose a trivialisa-

tion of the fibre Ep := E |p ∼= Cn, so

End(Ep

)⊗

OD|pΩ1X(D)

∣∣p∼= gln ⊗

CC = gln .


Thus, via this trivialisation of Ep, the restriction ∇|p determines a point in the Lie algebra gln. This

point is the residue of ∇ at p in this trivialisation of Ep. Changing the trivialisation of Ep moves

this point around in gln, and two trivialisations of Ep are related by conjugation by an element of

GL(Ep) ∼= GLn. So this point is constrained to an adjoint orbit op in gln. Thus, for every p ∈ D, the

restriction ∇|p determines an adjoint orbit op in gln.

Definition 2.25 (residue data)

Let (X,D) be a logarithmic pair. For every point p ∈ D, choose an adjoint orbit op in gln. The

collection of all these adjoint orbits,

oD :=op∣∣ p ∈ D

,

will be called residue data attached to D.

2. Connections with fixed residue data. Let D be reduced, and suppose we have two isomorphic

connections (E ,∇), (E ′,∇′) ∈ Conn(X,D, gln), and let φ : (E ,∇) → (E ′,∇′) be an isomorphism.

Then restricting to the divisor D, we find:

E E ⊗ Ω1X(D)

E ′ E ′ ⊗ Ω1X(D)

∇

φ φ⊗1

∇′

7−→

E∣∣D

E∣∣D⊗ Ω1

X(D)∣∣D

E ′∣∣D

E ′∣∣D⊗ Ω1

X(D)∣∣D

∇|D

φ|D φ|D⊗1

∇′|D

.

As in the previous paragraph, the horizontal maps in the commutative square on the right are

OD-linear. For every p ∈ D, upon choosing trivialisations of E |p, E ′|p, the restrictions ∇|p, ∇′|pdetermine points in adjoint orbits op, o

′p inside gln. The commutative square above on the right

says that these two points are related by conjugation by the element φ|p; in other words, these two

points lie in the same adjoint orbit, so in fact op = o′p. Therefore, the residue data is an invariant of

a connection.

Definition 2.26 (category of connections with fixed residue data)

We denote by Conn(X,D; gln, oD) the full subcategory of Conn(X,D; gln) consisting of gln-connections

on (X,D) with fixed residue data oD. We denote the corresponding moduli space by MdR(X,D; gln, oD).

In other words, if (E ,∇) ∈ Conn(X,D; gln, oD), then for every p ∈ D, the restriction ∇|p lies in the

adjoint orbit op ∈ oD.


3. Generic residue data. Let g be either the Lie algebra gln or sln. Fix a regular semisimple ele-

ment Λ ∈ g; this means that Λ is an n× n matrix which is diagonalisable with distinct eigenvalues.

We will say that Λ is generic if

(1) the eigenvalues of Λ have distinct real parts,

(2) no two eigenvalues of Λ differ by an integer.

We will say that an adjoint orbit o ⊂ g is a generic adjoint orbit if it is the adjoint orbit of a generic

regular semisimple element Λ ∈ g.

4. Existence of Levelt filtration.

Theorem 2.27 (existence of Levelt filtration)

Let (E ,∇) be a germ of a meromorphic connection of rank n at a point p ∈ X with a logarithmic

singularity at p. Assume that the residue Resp∇ is contained in a generic adjoint orbit. Then (E ,∇)

is equipped with a full Levelt filtration.

Proof.

Choose a local coordinate x vanishing at p. By theorem 2.22, there exists a trivialisation

E ∼−→ ⟨e1, . . . , en⟩Cx

in which ∇ has the diagonal form

∇ =loc d + Λdx

x,

where Λ = diag(λ1, . . . , λn) is the matrix of eigenvalues of the residue Resp∇, ordered by their

real parts as Re(λ1) < · · · < Re(λn). In this trivialisation, E has an obvious fundamental system of

∇-flat sections

Ψ(x) = x−Λ = x− diag(λ1,...,λn)

written in the basis of generators (e1, . . . , en). Denote by ψi := x−λiei the ∇-flat sections of E

corresponding to the columns of Ψ. The order on the eigenvalues λ1, . . . , λn implies an order by

growth rate at p on these ∇-flat sections: ψ1 ≺ · · · ≺ ψn, and therefore we obtain a ∇-invariant

filtration on E:

E• :=(0 ⊂ E1 ⊂ · · · ⊂ En := E

)where Ei := ⟨ψ1, . . . , ψi⟩OX,p

.


5. Nice connections with fixed residue data. Let (X,D) be a logarithmic pair, and let g be either

gln or sln. For each point p ∈ D, fix a generic adjoint orbit op ⊂ g. The collection oD :=op∣∣ p ∈ D

will be called generic residue data. The category Conn(X,D; g, oD) (definition 2.26) consists of

connections on (X,D) with fixed generic residue data oD. We make an additional genericity con-

straint on the the type of connections we consider in this thesis as follows. By theorem 2.27, any

connection (E ,∇) ∈ Conn(X,D; g, oD) is equipped for every p ∈ D with the Levelt filtration Ep• as-

sociated with p, which by definition is a filtration on E over a germ of a punctured neighbourhood

of p.

Definition 2.28 (nice connection)

A connection (E ,∇) on (X,D) is said to be nice if the following two conditions are satisfied:

(1) for every p ∈ D, the residue Resp∇ lies in a generic adjoint orbit of g;

(2) for every pair of points p, q ∈ D and any simply connected subset U ⊂ X containing p and q,

the corresponding Levelt filtrations Ep• , E

q• of E over U are transverse.

Levelt filtration is a very special kind of filtration, because it is respected by usual morphisms of

connections. That is, suppose (E ,∇), (E ′,∇′) ∈ Conn(X,D; g, oD) are two connections possessing

Levelt filtrations E•, E ′• associated with some p ∈ D. Then every morphism of connections φ :

(E ,∇) → (E ′,∇′) is necessarily Levelt filtration preserving. that is, the restriction of φ to a germ of

a punctured neighbourhood of p is a filtered morphism φ : Ep• → E ′p

• ; i.e., φ(Epi ) ⊂ E ′p

i for all i.

Definition 2.29 (category of nice connections with fixed residue data)

We denote by Conn∗(X,D; g, oD) the full subcategory of Conn(X,D; g, oD) whose objects are nice

meromorphic connections on (X,D) with residue data oD. We denote the corresponding moduli

space by M∗dR(X,D; g, oD).

Remark 2.30 (transversality of Levelt filtrations)

In definition 2.28, the condition of transversality between Levelt filtrations was stated in the Betti

picture: it is a constraint on the local system determined by the connection (E ,∇). Its equivalent

in the de Rham picture — i.e., as some kind of constraint on the residue data oD — is unknown to

us. It is possible that for some choice of oD, the category Conn∗(X,D; g, oD) is empty (or perhaps

uninteresting in some other sense). This situation, however, is non-generic (because transversality

is generic). Thus, by way of convention, whenever we say “fix some generic residue data oD


and consider the category Conn∗(X,D; g, oD)”, we implicitly assume that oD is chosen such that

Conn∗(X,D; g, oD) is nonempty.

6. Abelianisation of generic residue data. Let g be either the Lie algebra gln or sln. Let D ⊂ X

be a reduced divisor, and let oD be generic residue data in g. Then for each p ∈ D, the adjoint orbit

0p contains a diagonal element

Λp =

λp1 . . .λpn

,

whose eigenvalues λp1, · · · , λpn ∈ C have distinct real parts. Assume, without loss of generality, that

they are ordered by their real parts:

Re(λp1) < · · · < Re(λpn) .

Denote by oabp the ordered n-tuple of complex numbers (λp1, . . . , λ

pn). The collection of all such

n-tuples oabp will be referred to as the abelianisation of residue data oD:

oabD :=

oabp

∣∣ p ∈ D

.

One can think that each complex number λpi gives rise to an adjoint orbit in the 1-dimensional

abelian Lie algebra t ∼= C; this adjoint orbit is, of course, the number λpi itself, because t is abelian.

2.4. Linear Algebra of Transverse Flags

2.4.1. Flags and Filtered Transformations

Definition 2.31 (flag/filtration | filtered vector space)

A flag or filtration in an n-dimensional vector space E is an ascending chain of vector subspaces

E• :=(0 = E0 ⊂ E1 ⊂ · · · ⊂ En = E

),

such that dim(Ei) = i. A vector space E equipped with a flag E• is called a filtered vector space,

also denoted by E•.

What we call a flag is sometimes called a full or complete flag. Dropping the requirement that


dim(Ei) = i gives what is known as a partial flag.

Definition 2.32 (adapted basis)

Given a flag E• in E, a basis of (e1, . . . , en) of E is adapted to E• if Ei = ⟨e1, . . . , ei⟩.

If a flag E• is given by an adapted basis (e1, . . . , en), we will often denote E• simply by (e1, . . . , en).

Example 2.33 (standard flag)

The most basic example of a flag is the standard flag in Cn:

⟨e1⟩ ⊂ ⟨e1, e2⟩ ⊂ · · · ⊂ ⟨e1, . . . , en⟩

where (e1, . . . , en) is the standard basis for Cn.

Given a flag E•, one may consider the associated graded vector space

gr(E•) :=n⊕

i=1

Li where Li := Ei

/Ei−1 ..

Example 2.34 (associated graded of the standard flag)

For the standard flag (e1, . . . , en) on Cn, the i-th piece of the associated graded Li is generated by

the element ei mod ⟨e1, . . . , ei−1⟩.

Notice in this example that Li is of course naturally isomorphic to ⟨ei⟩, and hence to a subspace of

Cn. This is an artefact of a choice of an adapted basis for the flag in question. In general, without

an adapted basis, such an identification of a piece of the associated graded with a subspace of

E is not canonical. We will discuss ways to canonically identify pieces Li with subspaces of E in

section 2.4.2.

The natural notion of morphism on filtered vector spaces is linear transformations that preserve the

filtration.

Definition 2.35 (filtered unipotent transformation)

Let E• be a filtered vector space. A filtered transformation on E• is a linear transformation

G : E → E preserving the filtration E•, meaning that G(Ei) ⊂ Ei. A filtered transformation is

unipotent if it induces the identity transformation on the associated graded vector space gr(E•).


Example 2.36 (filtered transformation for the standard flag)

A filtered transformationG on Cn relative to the standard flag is (when represented in the standard

basis) any upper-triangular matrix; and it is unipotent if its diagonal entries are all 1:[∗ ∗ ∗0 ∗ ∗0 0 ∗

],

[1 ∗ ∗0 1 ∗0 0 1

].

2.4.2. Transverse Flags

We often need to assume a genericity condition on collections of flags, which takes the form of

transversality. Roughly, a pair of flags is transverse if the intersections of their subspaces all have

the expected dimension. Precisely:

Definition 2.37 (transverse flags)

Two flags Ep•,E

q• in an n-dimensional vector space E are transverse if

dim(Epi ∩ Eq

j ) = max0, i+ j − n .

If two flags Ep•,E

q• are transverse, we write Ep

• ⋔ Eq•.

Example 2.38 (pair of transverse flags)

The simplest example of a pair of transverse flags is the standard flag (e1, . . . , en) in Cn and its

opposite flag (en, . . . , e1).

1. A pair of transverse flags. Given a flag E• on vector space E, the individual pieces of the

associated graded vector space gr(E•) are not naturally identified with subspaces of E. However,

if E is endowed with a second flag that is transverse to E•, then the associated graded pieces are

indeed canonically identified with subspaces of E.

Let E be an n-dimensional vector space equipped with a pair of transverse flags:

Ep• :=

(0 = Ep

0 ⊂ Ep1 ⊂ · · · ⊂ Ep

n = E)

and Eq• :=

(0 = Eq

0 ⊂ Eq1 ⊂ · · · ⊂ Eq

n = E)

.

By transversality, their intersection determines a canonical decomposition of the vector space E into


a direct sum of lines:

E ∼−→ Ep• ∩ Eq

• :=n⊕

i=1

Epqi where Epq

i := Epi ∩ Eq

n+1−i .

Under this identification, we have the following isomorphisms:

Epi

∼−→ Epq1 ⊕ · · · ⊕ Epq

i−1 ⊕ Epqi ,

Epi−1

∼−→ Epq1 ⊕ · · · ⊕ Epq

i−1 .

We therefore obtain a canonical identification of the i-th piece Lpi of the associated graded gr(Ep•)

with the subspace Epqi of E. We highlight this simple observation in the following lemma:

Lemma 2.39

Given a pair of transverse flags Ep•,E

q•, there is a canonical identification of the associated graded piece

Lpidef= Ep

i

/Epi+1 with a subspace of E:

Lpi∼−→ Epq

i .

2.4.3. A Triple of Transverse Flags

1. Two different identifications of associated graded pieces. Now, let E be an n-dimensional

vector space equipped with three transverse flags:

Ep• :=

(0 = Ep

0 ⊂ Ep1 ⊂ · · · ⊂ Ep

n = E)

,

Eq• :=

(0 = Eq

0 ⊂ Eq1 ⊂ · · · ⊂ Eq

n = E)

,

Er• :=

(0 = Er

0 ⊂ Er1 ⊂ · · · ⊂ Er

n = E)

.

Consider the associated graded gr(Ep•). Using lemma 2.39, the individual graded pieces gr(Ep

•) can

be identified with subspaces of E in two different ways; namely,

Lpi∼−→ Epq

i and Lpi∼−→ Epr

i .

The resulting isomorphism Epqi

∼−→ Epri has a nice description. Consider the plane Epq

i ⊕ Epri . Unless

i = n, the expected dimension of the intersection Epi−1 ∩ (Epq

i ⊕ Epri ) would be 0; however, this

intersection is not transverse, because Epi−1 ⊂ Ep

i with codimension 1. (And if i = n, then the

intersection is transverse, so the expected dimension is 1.) At the same time, Epi−1 ∩ Eq

n+1−i = 0,


Epi−1 ∩ Er

n+1−i = 0 by transversality, so dim(Epi−1 ∩ (Epq

i ⊕ Epri )) = 1. Thus, Ep

i−1 intersects the plane

Epqi ⊕Epr

i in a line Epqr different from Epqi ,E

pri . The isomorphism Epq

i∼−→ Epr

i is the unique unipotent

filtered automorphism of the plane Epqi ⊕ Epr

i with respect to the flag Epqr ⊂ Epqi ⊕ Epr

i . We collect

this observation into the following lemma for future reference.

Lemma 2.40

The isomorphism Epqi

∼−→ Epri , defined by

Lpi

Epqi Epr

i ,

∼ ∼

∼

is induced by the unique unipotent filtered automorphism G of the plane Epqi ⊕ Epr

i with respect to the

flag

Epqr ⊂ Epqi ⊕ Epr

i where Epqr = Epi−1 ∩ (Epq

i ⊕ Epri ) .

Example 2.41

Let E = C3 equipped with the following three transverse flags:

Ep• :=

1

0

0

,0

1

0

,0

0

1

, Eq

• :=

0

0

1

,0

1

0

,1

0

0

, Er

• :=

1

2

1

,0

1

1

,0

0

1

.

Consider the second associated graded piece for the p-filtration: Lp2 = Ep2

/Ep1. It can be identified

with a subspace of E in two ways: Lp2∼−→ Epq

2 and Lp2∼−→ Epr

2 . In this case, Epq2 ⊕ Epr

2 = Ep2 and

Epqr = Ep1 ∩ (Epq

2 ⊕ Epr2 ) = Ep

1. Thus, the composite isomorphism Epq2

∼−→ Epr2 slides the line Epq

2 to

the line Epr2 inside the plane Ep

2 parallel to the line Ep1:


Ep1

Eq1

Er1

Eqr2

Epq2

Epr2

Ep2

2. A Triple of Transverse Flags. Now, let E be an n-dimensional vector space equipped with three

transverse flags:

Ep• :=

(0 = Ep

0 ⊂ Ep1 ⊂ · · · ⊂ Ep

n = E)

,

Eq• :=

(0 = Eq

0 ⊂ Eq1 ⊂ · · · ⊂ Eq

n = E)

,

Er• :=

(0 = Er

0 ⊂ Er1 ⊂ · · · ⊂ Er

n = E)

.

Consider the associated graded vector spaces corresponding to the flags Ep•,E

q•:

gr(Ep•) = Lp1 ⊕ Lp2 ⊕ · · · ⊕ Lpn where Lpi := Ep

i

/Epi−1 ,

gr(Eq•) = Lq1 ⊕ Lq2 ⊕ · · · ⊕ Lqn where Lqi := Eq

i

/Eqi−1 .

Proposition 2.42 (gluing isomorphisms)

For any pair of indices i, j, there is a canonical isomorphism φpqrij : Lpi

∼−→ Lqj .

Proof.

Since the flags Ep•,E

q• are transverse, we can canonically identify Lpi , L

qj with subspaces of E:

Lpi∼−→ Epq

i = Epi ∩ Eq

n+1−i and Lqj∼−→ Eqp

j = Eqj ∩ Ep

n+1−j .

Consider the plane Epqi ⊕ Eqp

j ⊂ E. The third flag Er• on E defines a third line Epqr ⊂ Epq

i ⊕ Eqpj that

is distinct from Epqi ,E

qpj as follows. The hyperplane Er

n−1 intersects the plane Epqi ⊕ Eqp

j in at least a


line (and at most a plane). By transversality, we have1:

dim(Epqi ∩ Er

n−1) = dim(Epi ∩ Eq

n+1−i ∩ Ern−1) = 0 .

Therefore, Epqi ⊂ Er

n−1. Similarly, Eqpj ⊂ Er

n−1, and so the line

Epqr := (Epqi ⊕ Eqp

j ) ∩ Ern−1

is distinct from Epqi ,E

qpj . Then by lemma 2.48, there exists a unique filtered unipotent automor-

phism G ∈ Aut(Epqr ⊂ Epqi ⊕ Eqp

j ) with the property that G(Epqi ) = Eqp

j . Then

φpqrij := G

∣∣Epqi

: Epqi

∼−→ Eqpj .

3. Examples of gluing isomorphisms. These gluing isomorphisms φpqrij play an essential role in

this thesis. Let us give some examples in low rank to illustrate their nature concretely by choosing

adapted bases. Throughout, we emphasise the calculation technique involved. The following is a

list of examples presented:

Example 2.43: rank 2 | Lp1∼−→ Lq2.

Example 2.44: rank 2 | Lq2∼−→ Lr1.




The computational technique described through these examples is later presented in full generality

of rank n on page 36.

Example 2.43

Consider the standard 2-dimensional vector space E := C2 equipped with three transverse flags:

Ep• :=

(0 ⊂ Ep

1 ⊂ E), Eq

• :=(0 ⊂ Eq

1 ⊂ E), Er

• :=(0 ⊂ Er

1 ⊂ E)

.

1This is because by mutual transversality, the codimension of the intersection is the sum of codimensions, and so thecodimension count gives codim(Ep

i ) + codim(Eqn+1−i) + codim(Er

n−1) = (n− i) + (n− n− 1 + i) + (n− n+ 1) = n


Denote the corresponding associated graded vector spaces as follows:

gr(E⋄•) := L⋄1 ⊕ L⋄2 where L⋄1 := E⋄

1 and L⋄2 := E/E⋄1.

(Here, ⋄ = p, q, r.) By transversality, we can always choose a basis (e1, e2) of E in which these flags

are

Ep• :=

10

,01

, Eq• :=

01

,10

, Er• :=

a1

,10

,

for some a ∈ C×. We introduce the following notation:

ep1 := e1 eq1 := e2 er1 := ae1 + e2

ep2 := e2 eq2 := e1 er2 := e1 .

So: E⋄• = (e⋄1, e

⋄2), L⋄1 = ⟨e⋄1⟩, L⋄2 =

⟨e⋄2 + E⋄

1

⟩.

To describe the isomorphism φpqr12 : Lp1

∼−→ Lq2, we first identify Lq2 with a subspace of E using

the pair of transverse flags Ep•,E

q•. The intersection of these flags gives a canonical decomposition

E ∼−→ Eq• ∩ Ep

• = Eq1 ⊕ Ep

1, which induces a canonical isomorphism

E/Eq1

∼−→ (Eq1 ⊕ Ep

1)/Eq1∼=can

Ep1 given by e1 + Eq

1 7−→ e1 .

Since Lq2 = E/Eq1 and Lq1 = Ep

1, the isomorphism φpqr12 is simply the inverse of this identification:

Lp1∼−→ Lq2 is given by ep1 7−→ eq2 + Eq

1 .

Notice that in this case φpqr12 is in fact independent of the filtration Er

•.

Example 2.44

We continue working in the setup and notation of example 2.43. To describe the isomorphism

φqrp21 : Lq2

∼−→ Lr1, we identify Lq2 with a subspace of E using the pair of transverse flags Eq•,E

r•. Their

intersection gives E ∼−→ Eq• ∩ Er

• = Eq1 ⊕ Er

1, which induces a canonical isomorphism

E/Eq1

∼−→ (Eq1 ⊕ Er

1)/Eq1∼=can

Er1 given by e1 + Eq

1 7−→ a−1(ae1 + a2) .

This formula is computed in the following manner: e1 + Eq1 = e1 + ⟨e2⟩ = e1 + a−1e2 + ⟨e2⟩ =


a−1(ae1 + e2) + ⟨e2⟩ 7→ a−1(ae1 + a2). Thus, since eq1 = e2, eq2 = e1, e

r1 = ae1 + e2, we have:

φqrp21 : Lq2

∼−→ Lr1 is given by eq2 + Eq1 7−→ a−1er1 .

Once again, φqrp21 is actually independent of the third filtration Ep

•.

Example 2.45

We continue working in the setup and notation of example 2.43. Let us describe the isomorphism

φpqr11 : Lp1

∼−→ Lq1. The spaces Lp1, Lq1 are already the subspaces Ep

1,Eq1 of E, so the isomorphism

φpqr11 is obtained as follows. The intersection Er

1 ∩ (Eq1 ⊕ Ep

1) of the plane Eq1 ⊕ Ep

1 (which is just

E) with the line Er1 is of course the line Er

1 itself. The isomorphism φpqr11 is then obtained as the

restriction G|Ep1

to Ep1 of the unique filtered unipotent automorphism G of E with respect to the

flag Er1 ⊂ E. This automorphism G is guaranteed by lemma 2.48 and is computed as follows. By

definition, G is filtered unipotent with respect to Er1 ⊂ E if it induces the identity grG = id on

gr(Er1 ⊂ E

)= Er

1 ⊕ E/Er1, which means that G

grG : er1 7−→ er1 and grG : ep1 + Er1 7−→ ep1 + Er

1 .

Let α ∈ C× be such that G(ep1) = αeq1; we use the filtered unipotent constraint to determine α:

ep1 + Er1

grG7−→ αeq1 + Er1 = αe2 + ⟨ae1 + e2⟩ = αe2 − α(ae1 + e2) + ⟨ae1 + e2⟩ = −αae1 + ⟨ae1 + e2⟩

= −αaep1 + Er1 .

Therefore, α = −a−1, and so G : ep1 7→ −a−1eq1. Thus:

φpqr11 : Lp1

∼−→ Lq1 is given by ep1 7−→ −a−1eq1 .

Example 2.46

We continue working in the setup and notation of example 2.43. To describe φpqr22 : Lp2

∼−→ Lq2, we

must first first identify Lp2 and Lq2 with subspaces of E using the pair of transverse flags Ep•,E

q•. The

intersection Ep• ∩Eq

• provides us with a decomposition E ∼−→ Ep1⊕Eq

1, which results in the following


canonical isomorphisms

Lp2∼−→ Eq

1 given by ep2 + Ep1 7−→ eq1 ,

Lq2∼−→ Ep

1 given by eq2 + Eq1 7−→ ep1 .

Then φpqr22 in these identification is described as follows. We intersect the plane Eq

1⊕Ep1 = E with Er

1

and compute the unique unipotent filtered automorphism G with respect to the flag Er1 ⊂ E. This

automorphism was very carefully described in example 2.45. We can find that G sends eq1 7→ −aep1,

and therefore:

φpqr22 : Lp2

∼−→ Lq2 is given by ep2 + Ep1 7−→ −aeq2 + Eq

1 .

Example 2.47

Consider the standard 3-dimensional vector space E := C2 equipped with three transverse flags:

Ep• :=

(0 ⊂ Ep

1 ⊂ Ep2 ⊂ Ep

3 =⊂ E), Eq

• :=(0 ⊂ Eq

1 ⊂ Eq2 ⊂ Eq

3 = E), Er

• :=(0 ⊂ Er

1 ⊂ Er2 ⊂ Er

3 = E)

.

Denote the corresponding associated graded vector spaces as follows:

gr(E⋄•) := L⋄1 ⊕ L⋄2 ⊕ L⋄3 where L⋄i := E⋄

i

/E⋄i−1.

(Here, ⋄ = p, q, r.) By transversality, we can always choose a basis (e1, e2, e3) of E in which these

flags are

Ep• :=

1

0

0

,0

1

0

,0

0

1

, Eq

• :=

0

0

1

,0

1

0

,1

0

0

, Er

• :=

a

b

1

,c

1

0

,1

0

0

,

for some a, b, c ∈ C. We introduce the following notation:

ep1 := e1 eq1 := e3 er1 := ae1 + be2 + e3

ep2 := e2 eq2 := e2 er2 := ce1 + e2

ep3 := e3 eq3 := e1 er3 := e1 .

So: E⋄• = (e⋄1, e

⋄2, e

⋄3), L⋄1 = ⟨e⋄1⟩, L⋄2 =

⟨e⋄2 + E⋄

1

⟩, L⋄3 =

⟨e⋄3 + E⋄

2

⟩.


To describe φpqr12 : Lp1

∼−→ Lq2, we begin by identifying Lq2 with a subspace of E using the pair

of transverse flags Ep•,E

q•. The intersection of these two flags gives a canonical decomposition

E ∼−→ Ep• ∩ Eq

• = Eqp1 ⊕ Eqp

2 ⊕ Eqp3 , which induces an isomorphism

Eq2

/Eq1

∼−→ Eqp2 = Eq

2 ∩ Ep2 .

Since Eq2 = ⟨e3, e2⟩ and Ep

2 = ⟨e1, e2⟩, we find Eq2 ∩ Ep

2 = ⟨e2⟩. Thus, since eq2 = e2, we have the

following identification:

Lq2∼−→ Eqp

2 = ⟨e2⟩ is given by eq2 + Eq1 7−→ e2 . (2.8)

Next, we must determine the intersection Ep1 ⊕ Eqp

2 ∩ Er2. Since Ep

1 = ⟨e1⟩,Eqp2 = ⟨e2⟩, and Er

2 =

⟨ae1 + be2 + e3, ce1 + e2⟩, this intersection is readily seen to be ⟨ce1 + e2⟩.

Thus, the desired isomorphism Ep1

∼−→ Eqp2 is the unique filtered unipotent automorphism of ⟨e1, e2⟩

with respect to the flag ⟨ce1 + e2⟩ ⊂ ⟨e1, e2⟩. After a short calculationa, we find that this automor-

phism sends e1 7→ −c−1e2. Combining this map with the identification (2.8), we find:

φpqr12 : Lp1

∼−→ Lq2 is given by ep1 7−→ −c−1eq2 + Eq1 .

aSuppose e1 7→ αe2. This map must satisfy e1 + ⟨ce1 + e2⟩ 7→ e1 + ⟨ce1 + e2⟩. Find α.

4. Gluing isomorphisms in general. In general, consider the standard n-dimensional vector

space E := C2 equipped with three transverse flags:

Ep• :=

(0 = Ep

0 ⊂ Ep1 ⊂ · · · ⊂ Ep

n = E)

,

Eq• :=

(0 = Eq

0 ⊂ Eq1 ⊂ · · · ⊂ Eq

n = E)

,

Er• :=

(0 = Er

0 ⊂ Er1 ⊂ · · · ⊂ Er

n = E)

.

Consider the corresponding associated graded vector spaces:

gr(E⋄•) := L⋄1 ⊕ L⋄2 ⊕ · · · ⊕ L⋄n where L⋄i := E⋄

i

/E⋄i−1 .


(Here, ⋄ = p, q, r.) By transversality, we can always choose a basis (e1, . . . , en) of E in which these

flags are

Ep• :=

1

0...

0

,0

1...

0

, . . . ,0

0...

1

, Eq• :=

0

0...

1

,0...

1

0

, . . . ,1

0...

0

, Er• :=

a11...

a1,n−2

a1,n−1

1

,

a21...

a2,n−2

1

0

, . . . ,

1

0...

0

0

,

for some aij ∈ C. We introduce the following notation:

epi := ei , eqi := en+1−i , eri :=n−i∑k=1

aikek + en+1−i ,

so: E⋄• = (e⋄1, . . . , e

⋄n), L⋄1 = ⟨e⋄1⟩, L⋄2 =

⟨e⋄2 + E⋄

1

⟩, · · · L⋄n =

⟨e⋄n + E⋄

n−1

⟩.

By proposition 2.42, there exists a canonical isomorphism φpqrij : Lpi

∼−→ Lqj . We now give an explicit

description of this isomorphism in the chosen adapted bases for Ep•,E

q•,E

r•. First, we identify the

spaces Lpi , Lqj with subspaces of E using the pair of transverse filtrations Ep

•,Eq•. Intersecting these

flags gives a canonical decomposition

E ∼−→ Ep• ∩ Eq

• = Epq1 ⊕ · · · ⊕ Epq

n = Eqp1 ⊕ · · · ⊕ Eqp

n ,

where Epqi

def= Epi ∩Eq

n+1−i. Since Epi = ⟨e1, . . . , ei⟩ and Eq

n+1−i = ⟨en, . . . , ei⟩, we find that Epqi = ⟨ei⟩;

and similarly, Eqpj = ⟨en+1−j⟩. Since Lpi =

⟨ei + Ep

i−1

⟩and Lqj =

⟨ej + Eq

j−1

⟩, we obtain the

following identifications:

Lpi∼−→ Epq

i given by epi + Epi−1 7−→ epi = ei ,

Lqj∼−→ Eqp

j given by eqj + Eqj−1 7−→ eqj = en+1−j .

If i = n+ 1− j, then the isomorphism Lpi∼−→ Lqj is given simply through these idenitificaitons:

Lpi∼−→ Lqj is given by epi + Ep

i−1 7−→ eqj + Eqj−1 (if i = n+ 1− j).

So we can now assume i = n + 1 − j, and it remains to describe the isomorphism Epqi

∼−→ Eqpj .


We consider the plane Epqi ⊕ Eqp

j in E, and we intersect it with the hyperplane Ern−1 ⊂ E. Since

Epqi ⊕Eqp

j = ⟨ei, en+1−j⟩ and this intersection is transverse, the resulting line in Epqi ⊕Eqp

j is spanned

by ei + αen+1−j for some constant α ∈ C×. To determine α, we express ei + αen+1−j as a linear

combination of the generators er1, . . . , ern−1 of Er

n−1:

ei + αen+1−j =

n−1∑k=1

Akerk =

(n−1∑m=1

Amam1

)e1 +

n−1∑k=2

(n−k∑m=1

Amamk +An+1−k

)ek +A1en , (2.9)

for someA1, . . . , An−1 ∈ C. This is a system of n linear equations in the n unknownsA1, . . . , An−1, α.

By transversality Ern−1 ⋔ Epq

i ⊕ Eqpj , it has a unique solution.

Thus, let α ∈ C× be the unique constant satisfying equation 2.9, so Ern−1∩E

pqi ⊕Eqp

j = ⟨ei + αen+1−j⟩.

Then the isomorphism Epqi

∼−→ Eqpj is the unique unipotent automorphism of the plane Epq

i ⊕ Eqpj =

⟨ei, en+1−j⟩ with respect to the flag ⟨ei + αen+1−j⟩ ⊂ ⟨ei, en+1−j⟩. To calculate this automorphism,

we look for β ∈ C× such that this automorphism sends ei 7→ βen+1−j . The condition of being

unipotent filtered means this automorphism is the identity on the associated graded vector space;

in practical terms, this means it must send the coset ei + ⟨ei + αen+1−j⟩ to itself. This allows us to

calculate β:

ei + ⟨ei + αen+1−j⟩ 7−→ βen+1−j + ⟨ei + αen+1−j⟩ = βen+1−j + ei + αen+1−j + ⟨ei + αen+1−j⟩ ,

whence β = −α. Therefore, the isomorphism Epqi

∼−→ Eqpj sends ei 7−→ −αen+1−j , and so we

conclude that

the isomorphism φpqrij : Lpi

∼−→ Lqj is given by epi + Epi−1 7−→ −αeqj + Eq

j−1 .

2.4.4. Transverse Flags in Two Dimensions

1. A pair of transverse flags. Let E be a 2-dimensional vector space, and let Lp, Lq ⊂ E be a pair

of distinct lines. Then E has a canonical decomposition E ∼−→ Lp ⊕ Lq, and this in turn provides

canonical identifications between the associated graded vector spaces gr(E•p), gr(E

•q) and E:

gr(E•p)

∼−→ Lp ⊕ Lq and gr(E•q)

∼−→ Lq ⊕ Lp .


2. A triple of transverse flags. Let E be a 2-dimensional vector space equipped with three trans-

verse flags:

Ep• :=

(0 ⊂ Ep

1 ⊂ E), Eq

• :=(0 ⊂ Eq

1 ⊂ E), Er

• :=(0 ⊂ Er

1 ⊂ E)

.

We have two canonical isomorphisms

E ∼−→ Eq• ∩ Ep

• = Eq1 ⊕ Ep

1 and E ∼−→ Eq• ∩ Er

• = Eq1 ⊕ Er

1 ,

and therefore we have a pair of natural isomorphisms:

E/Eq1

∼−→ Ep1 and E

/Eq1

∼−→ Er1 .

Thus, the triple of transverse filtrations Ep•,E

q•,E

r• provides a canonical isomorphism

Ep1

∼−→ Er1 .

Lemma 2.48

Let L1, L2, L3 be three distinct lines in a 2-dimensional vector space E, and consider the flag L2 ⊂ E.

There is a unique filtered unipotent transformation G ∈ Aut(L2 ⊂ E) with the property that

G∣∣L1

: L1 ∼−→ L3 .

Proof.

The vector space E has a canonical decomposition E = L1 ⊕ L2, which identifies the associated

graded gr(L2 ⊂ E) = L2 ⊕ L1. Choose generators as follows: L1 = ⟨e1⟩, L2 = ⟨e2⟩, and let α ∈ C

be such that L3 = ⟨e3 := e1 + αe2⟩. Then any filtered unipotent transformation G ∈ Aut(L2 ⊂ E) is

represented in the basis (e1, e2) by a lower-triangular matrix:

[G](e1,e2) =

[1 0

∗ 1

].

The requirement that G|L1 : L1 ∼−→ L3 forces ∗ to be α. This construction is obviously independent

of the chosen generators.


Geometrically, the transformation G in the proof

of lemma 2.48 projects L1 onto L3 along L2, as

shown on the right. By examining the explicit

construction ofG, we can give the following more

invariant meaning to the components of G, which

will be useful to us later.

L1

L2 L3

G

Lemma 2.49

The unique filtered unipotent transformation G ∈ Aut(L2 ⊂ E) sending L1 ∼−→ L3, constructed in

lemma 2.48, has the following form in the decomposition E = L1 ⊕ L2:

G =

[1 0

g 1

]:

L1 L1

L2 L2

⊕ ⊕

where the isomorphism g : L1 ∼−→ L2 factors through E/L3 up to a negative sign:

(g : L1 ∼−→ L2

)= −

(L1 ∼−→ E

/L3 ∼−→ L2

).

That is to say, there are natural identifications a : L1 ∼−→ E/L3, b : E

/L3 ∼−→ L2 (since L1, L2, L3 are

three distinct lines in E), and the claim regarding factorisation of g is that g = −b a.

Proof.

Continuing the notation of the proof of lemma 2.48, L1 = ⟨e1⟩, L2 = ⟨e2⟩, L3 = ⟨e3 := e1 + αe2⟩, so

g : e1 7→ αe2. At the same time, the composition L1 ∼−→a

E/L3 ∼−→

bL2 is computed as follows:

e1a7−→ e1 + L3 = e1 − (e1 + αe2) + L3 = −αe2 + L3

b7−→ −αe2 ,

where we used the fact that e1 + αe2 ∈ L3.

In a very similar vein, let us mention the following rather silly-looking factorisation of the identity

map. It will be useful to us when we discuss the Voros cocycle in later chapters.


Lemma 2.50

Given three distinct lines L1, L2, L3 in a 2-dimensional vector space E, the identity transformation

id : E → E can be written as

id =

[g13 0

g12 1

]:

L1 L3

L2 L2

⊕

g13

g12 ⊕ ,

where the isomorphisms g12 : L1 ∼−→ L2 and g13 : L1 ∼−→ L3 factor through E/L3 and E

/L2:

(g12 : L1 ∼−→ L2

)=(L1 ∼−→ E

/L3 ∼−→ L2

)and

(g13 : L1 ∼−→ L3

)=(L1 ∼−→ E

/L2 ∼−→ L3

).

That is to say, there are natural identifications a : L1 ∼−→ E/L3, b : E

/L3 ∼−→ L2 (since L1, L2, L3

are three distinct lines in E), and the claim is that g12 = −b a; and similarly for g13. The proof of

lemma 2.50 is similar to the proof of lemma 2.49.

chapter 3

Abelianisation

of sl2-Connections

Contents3.1 Quadratic Differentials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

3.1.1 Quadratic Differentials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

The space of quadratic differentials. . . . . . . . . . . . . . . . . . . . 44

Quadratic residue. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

Local normal form for quadratic differentials. . . . . . . . . . . . . . 46

3.1.2 Spectral Curves for Quadratic Differentials . . . . . . . . . . . . . . . . . . . 47

The log-cotangent bundle. . . . . . . . . . . . . . . . . . . . . . . . . 47

The spectral curve. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

The tautological 1-form. . . . . . . . . . . . . . . . . . . . . . . . . . 50

3.1.3 The Stokes Foliation F of the Spectral Curve . . . . . . . . . . . . . . . . . . 51

Foliation given by a 1-form. . . . . . . . . . . . . . . . . . . . . . . . 51

Foliation given by the tautological 1-form on Σ. . . . . . . . . . . . . 52

Local structure near a turning point. . . . . . . . . . . . . . . . . . . 52

Local structure near a pole. . . . . . . . . . . . . . . . . . . . . . . . 53

The Stokes graph. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

3.2 Cech Open Covers of Σ× and X× . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

3.2.1 The Levelt Open Cover of Σ× . . . . . . . . . . . . . . . . . . . . . . . . . . 55

Levelt cells. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

Involution-invariance. . . . . . . . . . . . . . . . . . . . . . . . . . . 56

Double and triple intersections. . . . . . . . . . . . . . . . . . . . . . 56

3.2.2 The Levelt Open Cover of X× . . . . . . . . . . . . . . . . . . . . . . . . . . 57

3.2.3 The Stokes Open Cover SΣ . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

Edge strips. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

Stokes half-regions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

Thickening the Stokes graph. . . . . . . . . . . . . . . . . . . . . . . 58

Stokes supersectors. . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

Intersection of Stokes supersectors. . . . . . . . . . . . . . . . . . . . 59

Stokes open cover. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

Double intersections in SΣ. . . . . . . . . . . . . . . . . . . . . . . . 61

3.2.4 The Stokes Open Cover of X× . . . . . . . . . . . . . . . . . . . . . . . . . . 61

Ideal triangulation of X. . . . . . . . . . . . . . . . . . . . . . . . . . 61

The Stokes open cover of X×. . . . . . . . . . . . . . . . . . . . . . . 62

3.3 The Abelianisation Functor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

42

TOC | chapter 3 | 43

3.3.1 Construction of (Eab,∇ab) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

Connections on the associated graded. . . . . . . . . . . . . . . . . . 64

Definition of the line bundle. . . . . . . . . . . . . . . . . . . . . . . 65

Gluing data. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

3.3.2 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

Canonical representatives for π∗Eab. . . . . . . . . . . . . . . . . . . 66

Functoriality. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

Oddness. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

Volume form. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

Monodromy around turning points. . . . . . . . . . . . . . . . . . . . 69

3.4 The Deabelianisation Functor . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

3.4.1 The Canonical Cocycle v(∇ab) . . . . . . . . . . . . . . . . . . . . . . . . . . 71

Canonical local identifications. . . . . . . . . . . . . . . . . . . . . . 71

Definition of the cocycle . . . . . . . . . . . . . . . . . . . . . . . . . 72

3.4.2 Properties of the Canonical Cocycle v(∇ab) . . . . . . . . . . . . . . . . . . 74

Local nature of the cocycle v(∇ab). . . . . . . . . . . . . . . . . . . . 74

The Voros matrices. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74

The detour path. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75

3.4.3 The Voros Cocycle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77

Universality of the canonical cocycle. . . . . . . . . . . . . . . . . . . 77

The Voros cocycle. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78

The deformed pushforward functor. . . . . . . . . . . . . . . . . . . . 79

The deabelianisation functor. . . . . . . . . . . . . . . . . . . . . . . 79

3.5 Abelianisation Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84

3.5.1 Abelianisation with respect to the Stokes Open Cover SX . . . . . . . . . . . 84

3.5.2 The Voros Class . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86

Deabelianisation functors. . . . . . . . . . . . . . . . . . . . . . . . . 86

Cech to sheaf cohomology. . . . . . . . . . . . . . . . . . . . . . . . . 86

The Voros class. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87

Unipotent cocycles. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87

3.5.3 Abelianisation Theorem. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89


3.1. Quadratic Differentials

The standard reference for a detailed study of quadratic differentials is Strebel’s book [Str84]. Also,

[MP98, section 4], [BS15, sections 2,3], [Sut14, chapter 2], [HN16, section 3], [GW16].

3.1.1. Quadratic Differentials

Let X be a smooth projective curve, and denote by ωX the canonical line bundle of X. If x is a local

coordinate on X, the line bundle ωX has a canonical local generator dx, called an adapted generator.

Definition 3.1 (meromorphic quadratic differential)

A meromorphic quadratic differential is a meromorphic section of the line bundle ω⊗2X .

Two quadratic differentials φ1, φ2 are equivalent if there is an automorphism F : X → X such that

F ∗φ1 = φ2. If x is a local coordinate on X, a quadratic differential φ is of the form

φ =loc q(x) dx⊗2 = q(x) dx⊗ dx ,

where q(x) is some meromorphic function on X. We usually write “dx2” instead of “dx⊗2”. If

z = z(x) is another local coordinate, then φ =loc q(z), where

q(x) = q(z(x)

)(dz

dx

)2

.

Using this transformation rule it can be checked directly that the zeroes and the poles of φ, as well

as their orders, are coordinate-independent notions. If a point p ∈ X is a zero or a pole of φ, it

is called a critical point of φ; otherwise, it is called a regular point. If a meromorphic quadratic

differential φ on X has poles bounded by a divisor D (effective, but not necessarily reduced), then

φ is a global holomorphic section of the line bundle ω2X(D); i.e., φ ∈ H0

(X, ω2

X(D)).

1. The space of quadratic differentials. We will be interested exclusively in quadratic differen-

tials which have poles of order exactly 2, so let D be a reduced effective divisor on X. The vector

space of meromorphic quadratic differentials on X with singularities along 2D is H0X

(ω2X(2D)

). By

the Riemann-Roch Theorem,

dimH0X

(ω2X(2D)

)= dimH1

X

(ω2X(2D)

)+ deg

(ω2X(2D)

)+ rank

(ω2X(2D)

)(1− gX) ,


where gX is the genus of X. If gX = 0, then ωX∼= OX(−2), so ω2

X(2D)∼= OX(−4 + 2|D|). Thus,

dimH0X

(ω2X(2D)

)=

2|D| − 3 for |D| ⩾ 2,

0 for |D| < 2.(gX = 0) (3.1)

If gX = 1, then ωX∼= OX, so deg

(ω2X(2D)

)= 2|D|. If D is nonempty, then dimH1

X(ω2X(2D)) = 0

by the Kodaira Vanishing Theorem. If D is empty, then dimH0X(ω

2X) = dimH0

X(OX) = gX = 1.

Therefore,

dimH0X

(ω2X(2D)

)=

2|D| for |D| > 0,

1 for |D| = 0.(gX = 1) (3.2)

Finally, if gX ⩾ 2, then the Riemann-Hurwitz Formula implies degωX = 2gX − 2, and the Kodaira

Vanishing Theorem implies dimH1X(ω

2X(2D)) = 0. Thus,

dimH0X

(ω2X(2D)

)= 2|D|+ 3gX − 3 . (gX ⩾ 2) (3.3)

2. Quadratic residue. In any local coordinate x centred at a point p ∈ X, a meromorphic

quadratic differential φ with a pole of order 2 at p is of the form

φ =( q0x2

+q1x

+ q2 + · · ·)dx2 ,

for some q0, q1, q2, . . . ∈ C. Clearly, the coefficient q0 is a coordinate-independent quantity; it is

called the quadratic residue of the quadratic differential of φ:

Res2p(φ) := q0 .

Thus, we have the following quadratic residue short exact sequence:

0 ω2X(D) ω2

X(2D) OD 0 .Res2

Since D is reduced, this sequence is indeed exact.


3. Local normal form for quadratic differentials. At a point p ∈ X which is neither a zero nor a

pole of φ, there is a distinguished local coordinate w in which

φ =loc dw2 .

The coordinate w is uniquely determined up to transformations of the form w 7→ ±w + const. If x

is an arbitrary local coordinate centred at p, and φ =loc q(x) dx2, then the distinguished coordinate

w is given by

w =

∫ √q(x) dx .

Since the point x = 0 is neither a zero nor a pole of q, the square root√q(x) is a well-defined

holomorphic function near x = 0, so this expression for w makes sense. For more details, see

[Str84, section 5.1].

Proposition 3.2 (local normal forms for quadratic differentials [Str84] )

Let φ be a meromorphic quadratic differential on X.

• If p ∈ X is a regular point of φ, then there is a distinguished local coordinate x centred at p in

which

φ =loc dx2 .

• If p ∈ X is a simple zero of φ, then there is a local coordinate x centred at p in which

φ =loc xdx2 .

• If p ∈ X is a second order pole of φ, and r ∈ C is the residue of φ at p, then there is a local

coordinate x centred at p in which

φ =loc rdx2

x2.

Definition 3.3 (nice quadratic differential)

A meromorphic quadratic differential φ ∈ H0(X, ω2

X(2D))

is nice if

• φ has only simple zeroes;

• φ has poles along 2D of order exactly 2;

• residues of φ are not real and negative or zero: Res2p(φ) ∈ R⩽0 for every p ∈ D.


A nice quadratic differential has exactly 12 degD+ gX − 1 zeroes.

3.1.2. Spectral Curves for Quadratic Differentials

1. The log-cotangent bundle. Let X be a smooth projective curve, and D ⊂ X be a reduced

effective divisor. Consider the logarithmic tangent sheaf AX := TX(− logD), and let a : AX → TX be

the inclusion map. Its dual A∨X = Ω1

X(logD) is the log-cotangent sheaf of (X,D). The log-cotangent

bundle is the total space Y := tot(A∨) def= tot(Ω1X(logD)

)with bundle projection map π : Y → X.

First, pullback AX to Y to obtain an inclusion π∗a : π∗AX → π∗TX of OY-modules. Second, since

π is a smooth morphism, the derivative π∗ : TY → π∗TX has full rank. Thus, we can form the fibre

product π!AX of π∗a and π∗:

π!AX π∗AX

TY π∗TX .

π∗a

π∗

The OY-module π!AX has a concrete description:

π!AX = TY(− log π∗D) .

The total space Y has a tautological meromorphic 1-form ηtaut ∈ π!A∨X = Ω1

Y(− log π∗D). In fact, Y

is log-symplectic.

2. The spectral curve. A quadratic differential φ is a section of ω⊗2X (2D), so consider the log-

cotangent bundle π : Y → X, where Y := tot(Ω1X(logD)

). Like the usual cotangent bundle, the

log-cotangent bundle has a tautological 1-form ηtaut ∈ H0Y(Ω

1Y(− log π∗D)).

Definition 3.4 (spectral curve of quadratic differential)

Let φ ∈ H0(X, ω⊗2

X (2D))

be a quadratic differential. The spectral curve of φ is defined as the zero

locus Σ in Y of the section η2taut − π∗φ:

Σ := Zero(η2taut − π∗φ) .

Denote by ι the inclusion map ι : Σ → Y. We continue to denote the restriction to Σ of the canonical


projection π : Y → X by π. Thus:Σ Y

X .

ι

ππ

If φ is a nice quadratic differential, then Σ is embedded in Y as a smooth divisor (see, for example,

[BNR89, section 3] and [DM14, section 4]). The spectral curve Σ of a nice quadratic differential φ

is simply ramified above the zeroes of φ:

Branch(π) = Zero(φ) ⊂ X .

Let us demonstrate these properties of Σ in a few instructive examples.

Example 3.5 (spectral curve near regular point)

Let φ be a nice quadratic differential on X with poles along 2D. Denote by B the set of zeroes of φ.

Let p ∈ X be a regular point of φ. Then using the local normal form, there is a distinguished local

coordinate (U, x) centred at p and disjoint from D ∪ B such that

φ =loc dx2 .

Over U, the log-cotangent bundle Y is isomorphic to the usual cotangent bundle T∨U = T∨

X|U, and

dx is the adapted generator. Let y denote the linear fibre coordinate on T∨U, so the tautological

1-form ηtaut equals y dx in this trivialisation of T∨U.

In these local coordinates, the spectral curve Σ of the quadratic differential φ is given as the affine

plane curve

Σ∣∣U=y2 = 1

⊂ A2

xy .

It has two disjoint components, and π : Σ|U → U is an unramified double cover.

Example 3.6 (spectral curve near a simple zero)

Let φ be a nice quadratic differential as in example 3.5, and let p be a simple zero of φ. Then using

the local normal form, we can find a local coordinate (U, x) centred at p and disjoint from D and

other points of B, in which

φ =loc x dx2 .

Again, the log-cotangent bundle Y is isomorphic to the usual cotangent bundle T∨U = T∨

X|U, and


dx is the adapted generator. Let y denote the linear fibre coordinate. Then the spectral curve Σ in

these local coordinates is the affine plane curve

Σ∣∣U=y2 = x

⊂ A2

xy .

It has a single smooth irreducible component, and the fibre coordinate can be used to parameterise

Σ|U:

Σ∣∣U=(z2, z) ∈ A2

xy

∣∣ z ∈ A1

.

In these coordinates, the projection π : Σ|U → U is the standard 2-fold covering map simply-

ramified over the origin:

π : z 7−→ z2 = x .

Example 3.7 (spectral curve near a second-order pole)

Let φ be a nice quadratic differential as in example 3.5, and let p be a second-order pole of φ.

Using the local normal form result, we can find a local coordinate (U, x) centred at p in which

φ =loc rdx2

x2,

where r is the residue of φ at p. The log-cotangent bundle Y = tot(Ω1X(D)

)is not isomorphic to the

usual cotangent bundle over U, but it still admits a simple description. Indeed, the divisor D locally

near p is simply cut out by the coordinate function x, so dx/x is a local generator of Ω1

X(D) near

p. Let y denote the linear fibre coordinate on Y|U, so the tautological 1-form ηtaut equals y dx/x

in this trivialisation.

In these local coordinates, the spectral curve Σ of the quadratic differential φ is given by as the

zero locus of the section

η2taut − π∗φ =

(ydx

x

)2

− rdx2

x2.

In other words, Σ|U is the affine plane curve

Σ∣∣U=y2 − r

⊂ A2

xy .

Thus, just as in example 3.5, Σ|U consists of two disjoint components, and the projection map

π : Σ|U → U is an unramified 2-fold cover.


3. The tautological 1-form. Since the spectral curve Σ embeds into Y via ι : Σ → Y as a smooth

divisor, the restriction

η := ι∗ηtaut

of the tautological 1-form to Σ is an abelian differential with simple poles along π∗D ⊂ Σ; in

symbols, η ∈ Ω1Σ(π

∗D). Since the spectral curve Σ is defined as the zero locus of the section

η2taut−π∗φ, the tautological 1-form η can be viewed as the distinguished square root of the quadratic

differential φ, which is the point of view taken in [DH75]. As a result, the tautological 1-form η has

simple zeroes precisely above the zeroes of the quadratic differential; i.e., η has simple zeroes only

along the ramification divisor:

Zero(η) = Ram(π) .

Definition 3.8 (turning point)

The zeroes of the tautological 1-form η ∈ Ω1Σ(π

∗D) will be called turning points. A zero of order

1 is known as a simple turning point.

Let us demonstrate these properties of η through a few examples.

Example 3.9 (tautological 1-form over a regular point)

In example 3.5, we found that Σ|U is given in local coordinates byy2 = 1

. So Σ|U has two

irreducible components y = 1 and y = −1. Therefore, the restriction η of the tautological

1-form ηtaut to Σ|U is very easy to deduce:

η =loc

dx on the component y = 1;

− dx on the component y = −1,

and the equality η2 = π∗φ is clearly seen.

Example 3.10 (tautological 1-form over a simple zero)

In example 3.6, it was established that Σ|U is given in local coordinates byy2 = x

, and we chose

the vertical fibre coordinate to parameterise Σ|U: Σ|U =(z2, z)

∣∣ z ∈ C

. Then the tautological

1-form ηtaut =loc y dx pulls back to Σ|U to

η =loc 2z2 dz .


At the same time,

π∗φ =loc

(2z2 dz

)2 ,

verifying the equality η2 = π∗φ.

Example 3.11 (tautological 1-form over a second-order pole)

In example 3.7, we found that Σ|U is given in local coordinates byy2 = r

, where r is the residue

of φ at the second-order pole p = x = 0. The tautological 1-form ηtaut =loc y dx /x pulls back to

Σ|U to

η =loc

√r dxx on the component y =

√r;

−√r dxx on the component y = −

√r,

where√r is any choice of a square root of r.

3.1.3. The Stokes Foliation F of the Spectral Curve

1. Foliation given by a 1-form. Let us first briefly describe in general how an abelian differential

on a complex curve determines a foliation. Let Σ be a smooth complex curve. If we view Σ as a real

surface, a local coordinate z can be written as z = x+iy where x, y are real. The local generator dz

of the holomorphic cotangent sheaf Ω1Σ is then expressed as dz = dx+ i dy. The real tangent sheaf

T RΣ is a sheaf locally generated by the coordinate vector fields ∂x, ∂y over the ring of C∞-functions

on Σ. Then we can consider paths in X along which the abelian differential dz is a real 1-form; that

is, we consider paths γ in X whose tangent vector γp ∈ TRpX at every point p ∈ γ satisfies

dz|p(γp) ∈ R .

Furthermore, the path γ is naturally oriented: we say that the tangent vector is γp ∈ TRpX is

oriented positively if dz|p(γp) > 0, and negatively if dz|p(γp) < 0. Thus, this construction yields a

distribution which integrates to a real 1-dimensional foliation; in the coordinate z, this distribution

consists simply of horizontal lines in the complex plane C.

More generally, an abelian differential η on a complex curve Σ naturally determines a singular

foliation on Σ (viewed as a real surface), whose leaves are paths in Σ along which η is a real

1-form. This foliation is singular precisely at the zeroes and poles of η.


2. Foliation given by the tautological 1-form on Σ. Now, let X be a complex projective curve, D

an effective divisor on X, and φ ∈ H0(X, ω⊗2

X (2D))

a nice quadratic differential. Let Σ be the spectra

curve of φ, and let η be the tautological 1-form on Σ. As an abelian differential on a complex curve,

η defines a singular foliation F on Σ. As the foliation determined by the distinguished square root

of a nice quadratic differential, the foliation F has further special properties (see [HM79] for more

details).

Definition 3.12 (Stokes-Levelt foliation, generic leaf, critical leaf)

The singular foliation of Σ determined by η will be called the Stokes-Levelt foliation and denoted

by F. A leaf of F is a critical leaf if its closure contains a turning point. Otherwise, a leaf is called

a generic leaf.

We may sometimes denote the collection of all critical leaves of F by Crit(F).

3. Local structure near a turning point. In example 3.10 we saw that the tautological 1-form

η ∈ Ω1Σ(π

∗D) near a simple turning point can be written in a local coordinate z as

η =loc 2z2 dz = 2

3 d(z3)

.

Writing z = x+ iy, we find that

η =loc (x2 − y2) dx− 2xy dy + i

((x2 − y2) dy + 2xy dx

).

The foliation F is the integration of the distribution ker(Im(η)

). So locally near z = 0, the leaves

are given as level sets of the integral of the 1-form (x2 − y2) dy + 2xy dx. As a result, they can be

explicitly and easily plotted1:

Y X

As is clear from this picture (or from the explicit algebraic expression presented above), there are

1To plot the level curves, we used the macOS software Grapher, version 2.5.


precisely six critical leaves of F passing through any simple turning point.

4. Local structure near a pole. To understand the local structure of the foliation F near a second-

order pole, recall from example 3.11 that the tautological 1-form η near a pole pi ∈ π∗D has the

form

η =loc adx

x,

where a ∈ C is one of two square roots of the residue r of φ at p. Writing x = reiθ, we find

η =loc a

(dr

r+ i dθ

).

So once again the integral curves of the distribution ker(Im(η)

)are easy to calculate and plot, but

their shape highly depends on the phase of the complex number a.

5. The Stokes graph.

Definition 3.13 (Stokes ray)

A critical leaf of F is called a Stokes ray if its closure contains a single turning point and a single

pole.

In symbols, a Stokes ray is a critical leaf ℓ such that ℓ \ ℓ consists of precisely two points, one of

which is a turning point and one of which is a pole.

Definition 3.14 (saddle connection)

A critical leaf of F is called a saddle connection if its closure contains two turning points.

Definition 3.15 (Stokes graph)

The collection of all critical leaves of F, turning points R, and poles π∗D will be called the Stokes

graph of φ and denoted by Γ.

Remark 3.16 (terminology: Stokes graph)

The terminology Stokes graph is borrowed from the exact WKB analysis literature, which itself drew

inspiration from similar terminology used in asymptotic analysis. The Stokes graph appearing in

WKB analysis is in some sense the global analogue of Stokes rays appearing in classical asymptotic

analysis of singular differential equations.


Let us make comment on the main difference between our terminology and what is found else-

where. The Stokes graph in the exact WKB literature is normally defined as the critical graph of

the quadratic differential, which is a graph on the base curve X. See, for instance, [IN14, definition

2.16], [KT05, definition 2.6]; see also [HN16, §3.1], [Vor83, equation (6.8), page 243], [DDP93].

The Stokes graph in the exact WKB literature, although defined on the base curve X, is implicitly

thought of as living on the spectral curve Σ. This is evidenced by the fact that many important

statements (e.g., the Voros theorem) in the exact WKB literature take into account the orientation

of the Stokes ray in question. Note that — as will be explained in more detail in section 3.2.4

— the notion of orientation for a Stokes ray only makes sense on the spectral curve Σ. Some au-

thors comment on this, see [IN14, §2.7]. Taking this into account, perhaps it is more appropriate

to interpret the definition of the Stokes graph in the exact WKB literature as the critical graph of

the quadratic differential together with the additional data of two orientations. In that case, our

definition is equivalent to this information.

Definition 3.17 (saddle-free quadratic differential)

A nice meromorphic quadratic differential φ on X is saddle-free if the Stokes graph Γ on Σ contains

no saddle connections.

Definition 3.18 (Stokes region)

A Stokes region is a quadrilateral on Σ formed by four edges of Γ which connect a pair of distinct

turning points to a pair of distinct poles.

A clarifying remark regarding this definition: a Stokes region U is an open subset of Σ, so its edges

are not included in U.

Proposition 3.19 (Stokes graph decomposition [BS15, Lemma 3.1] )

Let φ ∈ H0(X, ω2

X(2D))

be a nice saddle-free quadratic differential, and let Σ be its spectral curve.

Then its Stokes graph Γ decomposes Σ \ Γ into a finite number of Stokes regions.


3.2. Cech Open Covers of Σ× and X×

Let φ ∈ H0(X, ω2

X(2D))

be a nice saddle-free meromorphic quadratic differential, and let π : Σ → X

be its spectral curve. Let η be the canonical 1-form on Σ. Let R := Ram(π) ⊂ Σ be the ramification

divisor, and let B := Branch(π) ⊂ X be the branch locus. We puncture Σ at R and X at B:

Σ× := Σ \ R and X× := X \ B .

In this section, we define two convenient natural Cech open covers of Σ×, the Levelt cover (sec-

tion 3.3) and the Stokes cover (section 3.2.3). We then push both of these covers down to X× to

build two corresponding open covers of X×, which we also refer to as the Levelt cover (section 3.2.2)

and the Stokes cover (section 3.2.4).

3.2.1. The Levelt Open Cover of Σ×

First, we use the Stokes-Levelt foliation F on Σ to define a convenient Cech open cover U(Σ×) of

Σ×, which we call the Levelt open cover, consisting of Levelt open cells. This cover will be used

in section 3.3 to construct an important abelian connection on Σ×, a central object of study of this

thesis.

1. Levelt cells. The local structure of the foliation F near a pole pi ∈ π∗D shows that the collection

of all leaves incident to pi ∈ π∗D is an open set. Furthermore, recall that every point on Σ\(R∪π∗D)

lies on a leaf of the foliation F, and at least one of the endpoints of this leaf is a pole. This motivates

the following definition.

Definition 3.20 (Levelt cell, Levelt open cover)

For each pole pi ∈ π∗D, the Levelt cell associated with pi is defined as the open set

Upi :=

z ∈ Σ

∣∣∣ z lies on a leaf of F incident to pi∪ pi .

The collection

U :=Upi ⊂ Σ

∣∣∣ pi ∈ π∗D

.

forms an open cover of Σ×, called the Levelt open cover.

Open sets in U(Σ×) are enumerated by the poles pi ∈ π∗D, so there are precisely 2|D| Levelt cells


on Σ×. The boundary of each Levelt cell consists of turning points and poles joined by Stokes rays.

The following is a schematic illustrations of a few examples of typical Levelt cells. In each image,

the Levelt cell Upi is associated with the pole pi in the centre; the boundary ∂Up

i := Upi \ Up

i of

Upi is depicted in bright orange; the purple curves are some generic leaves joining a pole on the

boundary of Upi with the pole pi in the centre; faded orange rays in the middle are Stokes rays

joining a turning point on the boundary of Upi with the pole pi in the centre.

A note on notation: Recall that every point of π∗D is a preimage of a pole p ∈ D, and the two

preimages of p are denoted by p1, p2 ∈ π∗D (where the integer index refers to the order p1 ≺ p2

determined by the residue of η). So the notation “ Upi ” stands for “the Levelt cell associated with

the i-th preimage of p”.

2. Involution-invariance. The spectral curve π : Σ → X is equipped with a natural involution

automorphism σ : Σ → Σ. Recall that for any pole p ∈ D, the involution σ exchanges the two

preimages p1, p2 of p; in symbols, σ(p1) = p2. Recall furthermore that the Stokes-Levelt foliation F

is invariant under σ in the sense that any leaf ℓ of F incident to p1 is sent by σ to a leaf σ(ℓ) incident

to p2, and vice versa. As a result, for every pole p ∈ π∗D, the two Levelt cells Up1,U

p2 associated with

the two preimages p1, p2 of p are disjoint and are exchanged by σ; in symbols,

Up1 ∩ Up

2 = ∅ and σ(Up1) = Up

2 for all p ∈ D. (3.4)

3. Double and triple intersections. Every point z of Σ — with the exception of poles and turning

points — lies on a leaf ℓ of the foliation F. Such a leaf ℓ has two endpoints. If ℓ is a generic leaf (in

which case z is contained in a Stokes region), then these two endpoints are distinct poles in π∗D,

so z lies in two distinct Levelt cells. If ℓ is a Stokes ray, then only one of these endpoints is a pole in

π∗D, in which case z lies in a unique Levelt cell.


As a result, two Levelt cells Upi ,U

qj intersect in a disjoint union of Stokes regions2

Upqij := Up

i ∩ Uqj .

Since Stokes regions are disjoint, the Levelt open cover UΣ contains no triple intersections.

3.2.2. The Levelt Open Cover of X×

The Levelt open cover U(Σ×) of Σ× defines an open cover of X× simply by projecting every open

set in U(Σ×) to the base X× via the map π.

3.2.3. The Stokes Open Cover SΣ

Now we construct another useful Cech open cover of the punctured spectral curve Σ×, which we

call the Stokes open cover, SΣ. It consists of open sets which we call Stokes supersectors, as well as

small discs around the poles π∗D. We will use this cover to construct an important Cech 1-cocycle

in section 3.4.

The construction of a Stokes supersector is very simple: we split each Stokes region in half; then

each half is slightly enlarged along the edges. In this section, we carefully detail this procedure and

describe the double and triple intersections.

1. Edge strips. Consider a Stokes region U ⊂ Σ with pole vertices p1, q2 ∈ π∗D and turning point

vertices b1, b2 ∈ R, as shown schematically below on the left. Choose a pair of distinct generic leaves

e1, e2 ⊂ U joining q2 to p1, as shown below in the middle. The resulting open subset bounded by

e1, e2, highlighted below on the right, will be called an edge strip and denoted by eqp21.

p1

q2

b1 b2 e1 e2

p1

q2

b1 b2 e1 e2

p1

q2

Edge strips are enumerated by ordered pairs (q2, p1) of poles in π∗D such that q2, p1 are vertices of

a Stokes region. The order (q2, p1) is determined by the orientation of the Stokes-Levelt foliation

2In fact, the intersection of two Levelt cells is typically a single Stokes region.


F: the boundaries e1, e2 of eqp21 are generic leaves of F oriented from q2 to p1.

Furthermore, for every Stokes region U ⊂ Σ, there is another Stoke region U′ such that U,U′ are

related by the canonical involution σ; i.e., such that σ(U) = U′. We insist that the definition of edge

strips is involution-invariant. That is, we require that the choice of generic leaves e1, e2 in U and

e′1, e′2 in U′ is such that σ(ei) = e′i. It follows that

σ(eqp21) = epq21 . (3.5)

Let us make the following peripheral remark. Since η is an abelian differential without zeroes or

poles on U, its square η⊗2 is a metric on U. In this metric, an edge strip eqp21 is an actual horizontal

infinite strip in the complex plane C stretching from −∞ (which is the image of q2 in this metric)

to +∞ (which is the image of p1 in this metric). The idea is that the generic leaves e1, e2 are

chosen ‘close’ to each other, so that the edge strip eqp21 is a ‘narrow’ strip connecting q2 to p1 and

representing a ‘slight thickening’ of the an edge connecting q2 to p1. By means of the metric η⊗2,

these ‘narrowness’ requirements can be made into actual quantitative statements.

2. Stokes half-regions. A Stokes region U ⊂ Σ as in the previous paragraph containing an edge

strip eqp21 is a union of two open subsets, as show below. We will call these open sets Stokes half-

regions. Their intersection is an edge strip.

e1 e2

p1

q2

b1 b2 = e1 e2

p1

q2

b1 ∪ e2e1

q2

p1

b2

Thanks to (3.5), for every Stokes half-region V, there is another Stokes half-region V′ such that

σ(V) = V′.

3. Thickening the Stokes graph. Consider a Stokes half-region V as shown below on the left. We

choose two paths e′, e′′ contained in V and not intersecting the generic leaf e1, such that e′ joins p1

to b1, and e′′ joins q2 to b1, as shown below on the right.


e1 e2

ep1b1

eq2b1

p1

q2

b1e′

e′′e1 e2

ep1b1

eq2b1

p1

q2

b1

Notice that the paths e′, e′′ are not leaves of the Stokes-Levelt foliation F, because ep1b1 is the unique

leaf joining p1 to b1.

If V is another Stokes half-region such that σ(V) = V, then we insist on the choice of e′, e′′ such

that σ(e′) = e′ and σ(e′′) = e′′.

4. Stokes supersectors. The Stokes half-region V is adjacent to two other Stokes half-regions

V′,V′′, as shown below on the left. Let e′ be the path inside V′ joining p1 to b chosen as above;

and similarly let e′′ be the path inside V′′ joining q2 to b, as shown below in the middle. The open

set bounded by e2, e′, e′′, highlighted below on the right, will be called a Stokes supersector and

denoted by ∆qp21|b.

e2e2

p1

q2

r2

r1

b V

V′

V′′

e′

e′′

e2

p1

q2

r2

r1

b V

V′

V′′

e′

e′′

e2

p1

q2

r2

r1

b qp21|b

Stokes supersectors are enumerated by triples (q2, p1, b) consisting of a turning point b ∈ R and an

ordered pair (q2, p1) of poles in π∗D such that q2, p1, b are vertices of a Stokes region. The order

(q2, p1) is determined by the anti-clockwise order around the turning point b.

Every Stokes supersector ∆qp21|b has the shape of a triangular region: we will refer to e2, e

′, e′′ as the

edges of the Stokes supersector ∆qp21|b; and to b, q2, p1 as the vertices of ∆qp

21|b.


5. Intersection of Stokes supersectors. Two Stokes supersectors ∆,∆′ are called adjacent if they

intersect along an edge. Two adjacent Stokes supersectors ∆,∆′ may intersect in one of two ways.

If ∆,∆′ are both incident to the same turning point b ∈ R, then ∆,∆′ necessarily share a pole vertex

qi ∈ π∗D in common. We will call such an intersection open set a Stokes intersection and denote

it by ∆qi|b. The vertex qi is either a sink (in which case qi = q1 and we say ∆q

1|b is a sink Stokes

intersection) or a source (in which case qi = q2 and we say ∆q2|b is a source Stokes intersection).

Both cases are displayed below.

p2

q1

r2

bqp12|b

rq21|b

p1

q2

r1

bqp21|b

rq12|b

q1

b

q1|b := rq

21|b ∩ qp12|b

q2

b

q2|b := rq

12|b ∩ qp21|b

Stokes intersections are enumerated by a pair (b, qi) consisting of a turning point b ∈ R and a pole

qi ∈ π∗D. They are in one-to-one correspondence with edges of the Stokes graph.

If ∆,∆′ are adjacent Stokes supersectors that do not share a turning point vertex, then they must

intersect along an edge connecting two pole vertices pi, qj . Therefore, they intersect along an edge

strip:

p1

q2

b1 b2qp21|b1 pq

12|b2eqp21 = ∆qp

21|b1 ∩ ∆pq12|b2 .

6. Stokes open cover. Finally, put a small open disc neighbourhood Dpi of each pole pi ∈ π∗D

such that σ(Dp1) = Dp

2. The collection of all Stokes supersectors ∆pqij|b and disc neighbourhoods Dp

i

defines an open cover of Σ×, which we call the Stokes open cover and denote by SΣ.


7. Double intersections in SΣ. Double intersections in SΣ are of three types:

(1) Stokes intersection: intersection of two adjacent Stokes supersectors incident to the same

turning point;

(2) edge strip: intersection of two adjacent Stokes supersectors incident to different turning

points;

(3) intersection of a Stokes supersector and the disc neighbourhood of a pole.

Stokes intersections and edge strips we discussed in detail in paragraph 1 and paragraph 5. Let

us briefly describe the last type of double intersection. Let pi ∈ π∗D be a pole, Dpi ∈ SΣ a disc

neighbourhood of pi, and ∆pqij|b be a Stokes supersector for some qj ∈ π∗D. Then the double

intersection Dpi ∩ ∆pq

ij|b is a sectorial neighbourhood of pi with the same opening as ∆pqij|b:

qj

pi

b

pqij|b

Dpi

3.2.4. The Stokes Open Cover of X×

1. Ideal triangulation of X. Let us begin by making a few remarks that connect our constructions

with the literature. In the previous section, each Stokes region was split into tow Stokes half-

regions. Thanks to the local structure of the Stokes-Levelt foliation, each turning point has precisely

six distinct Stokes supersectors incident to it. Therefore, the union of all six of them forms a

punctured hexagonal neighbourhood of the turning point. Moreover, thanks to the condition (3.5)

we imposed on the edges, this hexagonal neighbourhood is invariant under the natural involution

σ : Σ → Σ. Thus, each hexagon is pushed down to X by π to a triangle. Each triangle has three

distinct poles in D as vertices and a single branch point in its face.

This collection of triangles on X forms an ideal triangulation of X. This triangulation is often called

the WKB triangulation and it has been used by several authors in a variety of applications. It

appears in [Sut14, end of section 3.2]; [BS15]; [GMN13b]; [IN14]; [FG06]; [HN16]; [BS15].


2. The Stokes open cover of X×. Each Stokes supersector ∆pqij|b ∈ SΣ has the property that

σ(∆pqij|b) = ∆pq

ji|b, and so we can define an open set

∆pqb := π(∆pq

ij|b) = π(∆pqji|b) .

p

q

b

Furthermore, for every p ∈ D, consider the disc neighbourhoods Dp1,D

p2, their images π(Dp

1), π(Dp2)

are disc neighbourhoods of p. Let

Dp := π(Dp1) ∩ π(D

p2) .

Clearly the collection of all open sets of the form ∆pqb and Dp forms an open cover of X×.

Definition 3.21 (Stokes open cover)

We shall refer to this open cover as the Stokes open cover of X×, denoted by SX.

3.3. The Abelianisation Functor

Let X be a smooth projective curve, and D a reduced effective divisor on X. Fix some generic

residue data oD in sl2. Explicitly in our case, this means that for each p ∈ D, we choose a number

λp ∈ C with positive real part, and we let op be the adjoint orbit in sl2 of diag(λp,−λp). Consider

the category Conn∗(X,D, sl2; oD) of nice connections with residue data oD (definition 2.29).

In this section, we define a certain functor with domain Conn∗(X,D, sl2; oD) whose codomain is a

certain category of abelian connections. This functor and its codomain depend on an additional

choice of a nice saddle-free quadratic differential φ ∈ H0(X, ω2

X(2D))

which is compatible with the

residue data in the following sense.


Definition 3.22 (compatibility φ and oD)

A quadratic differential φ ∈ H0(X, ω2

X(2D))

is said to be compatible with the residue data oD if

Res2D φ = det(ResD∇) for any ∇ ∈ Conn∗(X,D, sl2; oD).

Note that det(Resp∇) = −λ2p for every p ∈ D.

Given generic residue data and the fact that |D| ⩾ 3, such a quadratic differential can always be

chosen thanks to a theorem by Gupta and Wolf [GW16, Theorem 1.2]3.

In the previous section, from the data of φwe extracted the following information. Let π : Σ → X be

the spectral curve of φ. Recall that since φ is assumed to be nice, the spectral curve Σ is embedded

in the logarithmic cotangent bundle Y as a smooth divisor:

Σ Y := tot(Ω1X(logD)

)X .

ι

π π

The spectral curve is a 2-fold cover simply ramified over the branch locus B := Branch(π) ⊂ X of

π, which coincides with the zero locus of φ. Let R := Ram(π) ⊂ Σ be the ramification divisor of π,

and let Σ× := Σ \ R. The tautological 1-form η ∈ Ω1Σ(π

∗D) on Σ — defined as the pullback ι∗ηtaut

of the tautological 1-form ηtaut on Y — has simple zeroes precisely along R and simple poles along

π∗D. Every point p ∈ D has two preimages p+, p− ∈ π∗D on Σ, and the residues of η at p+, p− have

nonzero real parts of opposite signs. Let Resp± η =: ±λp where Re(λp) > 0.

We view the collection of complex numbers oabD :=

±λp

∣∣ p ∈ D

as the data of prescribed

residues that determines the category Conn∗(Σ×, π∗D; ab, 0abD ) of abelian meromorphic connections

on (Σ×, π∗D). This category is the codomain for the functor that we construct in this section.

3Thanks to Aaron Fenyes for pointing out this reference


Definition 3.23 (abelianisation functor)

Given a nice saddle-free quadratic differential φ ∈ H0(X, ω2

X(2D))

which is compatible with the

residue data oD, the functor

πab : Conn∗(X,D; sl2, oD) −→ Conn∗(Σ×, π∗D; ab, 0abD )

(E ,∇) 7−→ (Eab,∇ab) .

constructed in this section will be called the abelianisation functor. The abelian connection

(Eab,∇ab) will be called the abelianisation connection associated with (E ,∇) and φ.

This functor therefore defines a map M∗dR(X,D, sl2; oD) −→ M∗

dR(Σ×, π∗D, gl1; 0

abD ) between the

corresponding moduli spaces nice meromorphic connections with prescribed residues. We will in

fact show that the isomorphism class of the abelianisation connection [(Eab,∇ab)] is a point in the

de Rham-Prym variety DPrym∗π(Σ

×, π∗D, oabD ) ⊂ M∗dR(Σ

×, π∗D, gl1; 0abD ). Thus, the main result of

this section is the construction of the following map of moduli spaces:

πab : M∗dR(X,D, sl2; oD) −→ DPrym∗

π(Σ×, π∗D, oabD )

[(E ,∇)] 7−→ [(Eab,∇ab)] .

3.3.1. Construction of (Eab,∇ab)

Let a nice meromorphic connection (E ,∇) ∈ Conn∗(X,D, sl2; oD) and a nice saddle-free quadratic

differential φ ∈ H0(X, ω2

X(2D))

compatible with the residue data oD be given. We use the Levelt

open covering of Σ× to give a Cech construction of an abelian connection (Eab,∇ab) in the category

Conn∗(Σ×, π∗D, gl1; 0abD ).

1. Connections on the associated graded. Let p ∈ D, and let Up1,U

p2 be the Levelt cells associated

with the two preimages p1, p2 ∈ π∗D of p. Let Up := π(Up1) = π(Up

2) ⊂ X; it is a simply connected

open neighbourhood of p. By theorem 2.27, the connection (E ,∇) is naturally equipped over Up

with the Levelt filtration corresponding to p:

Ep• :=

(0 ⊂ Lp

1 ⊂ E)

.


Consider the associated graded bundle, defined over Up:

gr(Ep• ) = Lp

1 ⊕ Lp2 where Lp

2 := E/

Lp1 .

Since the Levelt line Lp1 is ∇-invariant, ∇ induces rank-1 connections ∇p

1,∇p2 on each Lp

1,Lp2:

∇p1 : Lp

1 −→ Lp1 ⊗ Ω1

U(p) and ∇p2 : Lp

2 −→ Lp2 ⊗ Ω1

U(p) .

Indeed, ∇p1 is obviously just the restriction of ∇ to Lp

1. The connection ∇p2 is defined as follows.

Given a section e ∈ Lp2 = E

/Lp1, let e be any lift of it to E , so ∇(e) is a section of E ⊗ Ω1

U(p). Then

∇p2(e) is defined as the projection of ∇(e) to Lp

2 ⊗ Ω1U/S(p). Since any other lift of e to E differs

from e only by a section of Lp1 — which itself is a ∇-invariant subsheaf, — the image ∇p

2(e) is

independent of the choice of a lift of e to E .

2. Definition of the line bundle. Since π : Upi → Up is an isomorphism, the connection (Lp

i ,∇pi )

is naturally (via pullback by the isomorphism π|Upi) a rank-1 meromorphic connection on Up

i ⊂ Σ

with at most a regular singularity at pi ∈ π∗D. Thus, for every Upi ∈ U, we define:

(Eab,∇ab)∣∣Upi

:= (Lpi ,∇

pi ) .

All that remains is to describe the gluing data over all double intersections.

3. Gluing data. All double intersections in the Levelt open covering U are disjoint unions of

Stokes regions. So let U be a Stokes region. The vertices of U are a pair of turning points and a

pair of distinct poles pi, qj ∈ π∗D, one of which is a sink and one of which is a source. Without

loss of generality, assume assume that pi is a source and qj is a sink (so i = 2, j = 1). Thus,

U is (a connected component of) the double intersection Upi ∩ Uq

j . We define a canonical gluing

isomorphism gpq21 : Lp2

∼−→ Lq1 intertwining the connections ∇p

2,∇q1 as follows.

The Stokes region U is incident to both p2 and q1, so the restriction4 of π∗E |U = E |U is naturally

equipped with two Levelt filtrations Ep• , E

q• , which are transverse since ∇ is nice. By transversality,

we have a canonical decomposition E ∼−→ Lp1⊕Lq

1. This decomposition induces the desired isomor-

phism gpq21 : Lp2 = E

/Lp1

∼−→ Lq1. Finally, the isomorphism gpq21 obviously intertwines the connections

∇p2,∇

q1 because they are induced by the same connection ∇.

4notice that π restricted to U is an isomorphism onto its image, so π∗E |U is canonically isomorphic (via π) to E |U.


The Levelt open covering U contains no triple intersections so the construction of (Eab,∇ab) over

Σ× is complete.

3.3.2. Properties

1. Canonical representatives for π∗Eab. Recall the Levelt open cover U(X×) of X× obtained by

pushing down via π the Levelt open cover U(Σ×) of Σ×. If p ∈ D, then the open set Up ∈ U(X×)

has two disjoint preimages Up1,U

p2 ∈ U(Σ×) which are Levelt cells corresponding to the poles p1, p2.

The line bundle Eab is canonically represented over Upi by the line bundle Lp

i , which is a piece of

the associated graded bundle gr(Ep• ). Therefore, the vector bundle π∗Eab is canonically represented

over each open set Up ⊂ X by the associated graded vector bundle gr(Ep• ).

2. Functoriality. Fix generic residue data oD ⊂ sl2, and a quadratic differential φ compatible with

oD. Let π : Σ → X be the spectral curve of φ.

Lemma 3.24

The correspondence (E ,∇) 7−→ (Eab,∇ab) extends to a functor

πab : Conn∗(X,D, sl2; oD) −→ Conn∗(Σ×, π∗D, gl1; 0abD ) .

Proof.

Recall that a morphism (E ,∇) → (E ′,∇′) of two meromorphic connections on (X,D) in the cat-

egory Conn∗(X,D, sl2; oD) is an OX-linear map φ : E → E ′ which intertwines ∇,∇′ with the

additional property that for each p ∈ D, it is a filtered transformation Ep• → E ′p

• . Thus, φ in-

duces a diagonal transformation on the associated graded bundles gr(Ep• ) → gr(E ′p

• ) which inter-

twines the abelian connections on the graded pieces. Since (Eab,∇ab) and (E ′ab,∇′ab) were con-

structed precisely out of these associated graded pieces with connections, φ induces a morphism

(Eab,∇ab) → (E ′ab,∇′ab).

3. Oddness. The fact that (π∗Eab, π∗∇ab) is a connection on the punctured curve X× = X \ B

implies immediately that (π∗Eab, π∗∇ab) is an sl2-connection. Indeed, since X× is a non-compact

curve, any line bundle with connection on (X×,D) is trivialisable to the canonical module(OX×(−D),∇D

),

hence so is(det(π∗Eab), tr(π∗∇ab)

). In fact, we can make a more careful statement as follows.


Proposition 3.25

Let (E ,∇) be a nice meromorphic sl2-connection, and φ be a nice saddle-free quadratic differential

compatible with the residue data oD of ∇. Let (Eab,∇ab) be the corresponding abelianisation connec-

tion. Then there is a canonical isomorphism

det(π∗Eab) ∼−→ det E

which intertwines the connections tr(π∗∇ab) and tr∇. In particular, the isomorphism class of (Eab,∇ab)

defines a point in the de Rham-Prym variety:

[(Eab,∇ab)] ∈ DPrym(π, oabD ) ⊂ M∗dR(Σ

×, π∗D, gl1; 0abD ) .

In other words, the abelianisation connection (Eab,∇ab) corresponding to an sl2-connection (E ,∇)

is necessarily an odd connection.

Proof.

First, a general fact: if E is a rank-2 vector bundle which is an extension of line bundles,

0 L E L′ 0 ,

then the determinant line bundle det E is naturally isomorphic to L ⊗ L′.

Now, consider the Levelt open cover U(X×) of X×. If p ∈ D, then over the Levelt cell Up ⊂ X, the

vector bundle E is endowed with the Levelt filtration Ep• = (Lp

1 ⊂ E). So E is naturally an extension

of line bundles over Up:

0 Lp1 E Lp

2 0 ,

where Lp2

def= E/

Lp1. Therefore, the determinant line bundle det(E) over Up is naturally isomorphic

to the line bundle Lp1 ⊗ Lp

2. At the same time, the pushforward vector bundle π∗Eab over Up is

naturally represented by the associated graded gr(Ep• ) = Lp

1 ⊕ Lp2, so det(π∗Eab) is also naturally

identified with the line bundle Lp1 ⊗ Lp

2. It follows that the line bundles det(E) and det(π∗Eab) are

naturally isomorphic.


4. Volume form. If (E ,∇) is an sl2-connection on (X,D), then by definition det E is trivialisable

to the canonical module(OX(−D),∇D

). We show that a choice of this trivialisation is equivalent to

equipping the abelian connection (Eab,∇ab) with an additional structure, which we now deduce.

Let µ : (det E , tr∇) ∼−→(OX(−D),∇D

)be the chosen trivialisation. Over any Levelt cell Up ⊂ X,

the vector bundle E is naturally filtered by the Levelt filtration Ep• . At the same time, the vector

bundle π∗Eab is canonically represented by Lp1 ⊕ Lp

2 over Up. Then by proposition 3.25, det E is

canonically identified with det(Lp1 ⊕ Lp

2) = Lp1 ⊗ Lp

2. Therefore, the isomorphism µ induces an

isomorphism

µ : Lp1 ⊗ Lp

2∼−→ O(−D) (3.6)

over Up denoted by the same letter. Due to the anti-symmetry of the exterior product in the defini-

tion of det E , this map µ satisfies the following anti-symmetry property:

µ(s1, s2) = −µ(s2, s1) . (3.7)

Using the fact that the projection π restricts to an isomorphism Upi → Up for each i, we have the


Lp1 ⊗ Lp

2∼−→ (Eab ⊗ σ∗Eab)

∣∣Up1

and Lp1 ⊗ Lp

2∼−→ (σ∗Eab ⊗ Eab)

∣∣Up2

.

Via these identifications, the isomorphism (3.6) induces a ∇ab-flat isomorphism

µ : Eab ⊗ σ∗Eab ∼−→ OΣ×(−π∗D) . (3.8)

Said differently, we obtain a ∇ab-flat map µ : Eab → σ∗(Eab)∨ which is an isomorphism over Σ×

but drops rank along π∗D. Pulling µ : Eab ∼−→ σ∗(Eab)∨ back by σ, dualising, and using the identity

σ2 = id, we obtain another isomorphism σ∗µ∨ : Eab → σ∗(Eab)∨ which likewise drops rank along

π∗D because σ(π∗D) = π∗D. Then the identity (3.7) implies

σ∗µ∨ = −µ . (3.9)

Thus, a trivialisation µ : (det E , tr∇) ∼−→(OX(−D),∇D

)induces a ∇ab-flat isomorphism (3.8) satis-

fying (3.9). Conversely, the isomorphism (3.8) induces an isomorphism det(π∗Eab) ∼−→ OX×(−D),

which by proposition 3.25 induces a ∇-flat isomorphism det E ∼−→ OX×(−D). Since ∇ is smooth


along the branch locus B, this isomorphism extends to an isomorphism det E ∼−→ OX(−D). This

establishes the claim.

Remark 3.26 (Equivariance)

This additional structure µ on Eab has appeared previously in [HN16, equation (4.1)], where it

is called an equivariant structure. Indeed, by restricting the isomorphism µ in (3.8) to the locus

Σ× \ π∗D, we recover the isomorphism (4.1) in [HN16] precisely, and equation (4.4) in [HN16] is

readily seen to be equivalent to equation (3.9) above.

5. Monodromy around turning points.

Lemma 3.27

The connection (L,∇ab) over Σ× has monodromy −1 around every turning point.

Proof.

Let a ∈ R be a turning point. Then a is a vertex of six Levelt cells as displayed:

Up1

Uq2Ur

1

Up2

Uq1 Ur

2

Thus, the computation of the monodromy of ∇ab around the turning point a amounts to computing


the composition of the transition functions in the corresponding order:

gpq12

gqr21

grp12

gpq21

gqr12

grp21

Lp1

Lq2Lr

1

Lp2

Lq1 Lr

2

Lr1 Lq

2

Lp1 Lp

2

Lq1 Lr

2

∼

grp12

∼gqr21

∼

gpq21

∼

gpq12

∼gqr21

∼

grp12

.

A simple computation shows immediately that this composition equals −1.

3.4. The Deabelianisation Functor

Consider the pushforward sl2-connection (π∗Eab, π∗∇ab) on (X×,D). A natural question is: what is

the relationship between the original connection (E ,∇) and (π∗Eab, π∗∇ab)? The most important

and immediate observation is that these connections are certainly not isomorphic: the original con-

nection ∇ is smooth along the branch locus B, whereas π∗∇ab has nontrivial monodromy around

every point of B which means (π∗Eab, π∗∇ab) does not admit a holomorphic extension over the

branch locus.

Nevertheless, there is a natural way to compare them. Given the explicit construction of ∇ab in

the previous section, we are able to construct canonical local identifications in the Stokes cover SX

between E and π∗Eab. We then use these identifications to construct a Cech 1-cocycle with respect

to the cover SX that encodes the difference between (E ,∇) and (π∗Eab, π∗∇ab). Importantly, this

cocycle is also canonical. This is the content of proposition 3.28.

Investigating this canonical cocycle further we find that it is local to the branch locus and appears

in a standard form that is completely impartial to the particular abelian connection selected on

the spectral curve Σ×. Our mathematical interpretation of these observations is that the canonical

cocycle can be viewed as a Cech 1-cocycle with respect to SX with values in the sheaf Aut(π∗) of

natural automorphisms of the pushforward functor π∗.

Throughout this section, let (E ,∇) be a nice sl2-connection on (X,D), and let q ∈ H0(X, ω2

X(2D))

be a nice saddle-free quadratic differential which is compatible with the residue data of ∇. Let


π : Σ → X be the spectral curve determined by q. Let B := Branch(π) ⊂ X and R := Ram(π) ⊂ Σ

be the branch and ramification loci of π, and let X× := X \ B, Σ× := Σ \ R. Let (Eab,∇ab) be the

abelianisation connection on (Σ×, π∗D) associated with ∇ and q.

3.4.1. The Canonical Cocycle v(∇ab)

In this subsection, we prove the following proposition.

Proposition 3.28

Given a nice meromorphic connection (E ,∇) over (X,D) and a nice saddle-free quadratic differential

compatible with the residue data of ∇, let (Eab,∇ab) be the abelianisation connection. Then there is

a canonical π∗∇ab-flat Cech 1-cocycle v(∇ab) with respect to the Stokes open cover SX with values in

the sheaf of automorphisms of π∗Eab,

v(∇ab) ∈ Z1(SX,Aut(π∗Eab)

),

which acts on the connection (π∗Eab, π∗∇ab) to produce (E ,∇).

The main significance of this proposition is the canonicity of v(∇ab). Normally, any two vector

bundles differ by a cocycle with respect to any open cover, but only the cohomology class of this

cocycle is canonical. The point of proposition 3.28 is that there exists a Cech cover SX of X× and a

cocycle with respect to SX which is defined canonically, not just up to a coboundary.

Proof of proposition 3.28.

1. Canonical local identifications. Let ∆pqb ∈ SX be a Stokes supersector, where p, q ∈ D and

b ∈ B:

p

q

b


The open set ∆pqb is incident to both poles p and q, so the pushforward vector bundle π∗Eab has two

natural presentations over ∆pqb : as the associated graded bundle for either the Levelt filtration Ep

•

or the Levelt filtration Eq• :

π∗Eab∣∣∆pqb

∼−→ gr(E⋄• )∣∣∆pqb

(⋄ = p, q).

At the same time, by the transversality of Ep• , E

q• , the vector bundle E is naturally identified with the

corresponding associated graded bundles:

E ∼−→ gr(E⋄• ) (⋄ = p, q).

Explicitly, over ∆pqb , the vector bundle E has a canonical decomposition E ∼−→ Lq

1 ⊕ Lp1 coming

from the intersection Eq• ⋔ Ep

• . In this decomposition, the map E ∼−→ gr(Eq• ), for example, can be

expressed in matrix form as follows:

(φpqb : E

∣∣∆pqb

∼−→ gr(Eq• )∣∣∆pqb

)=

[1 0

0 g

]:

Lq1 Lq

1

Lp1 Lq

2

⊕ ⊕

g

, (3.10)

where g : Lp1

∼−→ Lq2 is the canonical isomorphism Lp

1∼−→(Lq1 ⊕ Lp

1

)/Lq1.

2. Definition of the cocycle We now define the 1-cocycle v(∇ab) with respect to SX. We need to

provide an automorphism of π∗Eab over each double intersection in SX. Recall that that there are

two main types of double intersections, and that the second type comes in three flavours.

(1) Consider first a Stokes intersection. Let ∆pqb ,∆

qrb be two adjacent Stokes supersectors that are

incident to the same branch point b, and consider their intersection ∆qb = ∆pq

b ∩∆qrb , as shown:

p

q

r

b

The double intersection ∆qb is contained in the Stokes supersectors ∆pq

b ,∆qrb , so we have the


following two canonical identifications over ∆qb:

φpqb : E ∼−→ gr(Eq

• ) and φqrb : E ∼−→ gr(Eq

• ) . (3.11)

At the same time, ∆qb is incident to the pole q, so the associated graded vector bundle gr(Eq

• )

canonically represents π∗Eab over ∆qb. Composing these identifications, we obtain a canonical

automorphism φqrb (φpq

b )−1 of π∗Eab over ∆qb. Therefore, we define v(∇ab) on ∆q

b to be the

automorphism

vqb(∇ab) := v(∇ab)

∣∣∆qb

:= φqrb (φpq

b )−1 ∈ Aut

(π∗Eab∣∣

∆qb

). (3.12)

This definition takes into account the convention of respecting the anti-clockwise order of

vertices (p, q, r) of the triangle ∆b.

(2) Next, we consider an edge strip. Let ∆pqb1,∆qp

b2be two adjacent Stokes supersectors which are

incident to two different branch points b1, b2, and consider their double intersection epq :=

∆pqb1

∩ ∆qpb2

.

This double intersection is incident to two poles p, q, so π∗Eab is canonically represented over

epq in two different ways: as gr(Ep• ) and as gr(Eq

• ). At the same time, the open sets ∆pqb1,∆qp

b2

are both incident to the same two poles p, q. This means that the two identifications φpqb1, φqp

b2

of E with either gr(Eq• ) or with gr(Ep

• ) coincide over epq. So their composition is the identity

on π∗Eab. We define v(∇ab) on epq to be the identity automorphism:

v(∇ab)∣∣epq

:= φqpb1

(φqpb2)−1 = id ∈ Aut

(π∗Eab∣∣

epq

). (3.13)

The remaining double intersections are dealt with similarly. We have thus defined automorphisms


of π∗Eab over each double intersection in SX. All that remains to complete the proof of proposi-

tion 3.28 is to check that these automorphisms satisfy the cocycle condition, which is easy.

3.4.2. Properties of the Canonical Cocycle v(∇ab)

The canonical cocycle v(∇ab), by its very definition, reconstructs (E ,∇) from (Eab,∇ab), and that is

its main property. In this subsection, we investigate the structure of this canonical cocycle further.

1. Local nature of the cocycle v(∇ab). By examining the definition of the cocycle v(∇ab) in

the proof of proposition 3.28, we notice that it is fully determined by its values on the double

intersections of type 1. This means that the cocycle v(∇ab) is depends only its values in triangular

open neighbourhoods of the branch locus. We refer to this property of v(∇ab) as being local to each

branch point.

2. The Voros matrices. Let us now compute the cocycle v(∇ab) explicitly. Due to the local nature

of the cocycle, we need only to compute it on double intersections of type 1. Let us therefore return

to (3.14), which is:

vqb(∇ab) def= φqr

b (φpqb )−1 . (3.14)

This is an automorphism of the vector bundle π∗Eab over the double intersection ∆qb = ∆pq

b ∩ ∆qrb .

Over ∆qb, the vector bundle π∗Eab is canonically represented by gr(Eq

• ), and the maps φqrb and φpq

b

are the canonical identifications between E and gr(Eq• ). Using (3.10), they can be expressed in

matrix form as follows:

(φqrb : E ∼−→ gr(Eq

• ))=

[1 0

0 h2

]:

Lq1 Lq

1

Lr1 Lq

2

⊕ ⊕ ;

(φpqb : E ∼−→ gr(Eq

• ))=

[1 0

0 h1

]:

Lq1 Lq

1

Lp1 Lq

2

⊕ ⊕

h1

.

Then the cocycle vqb(∇ab) is computed as follows:

vqb(∇ab) = φqr

b (φpqb )−1 =

[1 0

0 h2

]·

[1 I10 I2

]·

[1 0

0 h−11

]:

Lq1 Lq

1 Lq1 Lq

1

Lq2 Lp

1 Lr1 Lq

2

⊕ ⊕ ⊕ ⊕

h−11

I2

I1

h2

,(3.15)


where the matrix in the middle is the identity id : E → E written as a map Lq1 ⊕ Lp

1 → Lq1 ⊕ Lr

1,

and where I1, I2 are some maps which are will describe shortly. First, using lemma 2.50, we see

that the map I2 : Lp1 → Lr

1 factors through Lq2 as h−1

2 h1. Therefore, we find that vqb is a unipotent

upper-triangular matrix:

vqb(∇ab) =

[1 ∆

0 1

]:

Lq1 Lq

1

Lq2 Lq

2

⊕ ⊕∆ , (3.16)

where ∆ := I1h−11 . We collect the result of this calculation into the following lemma. Recall that

the vector bundle π∗Eab in any simply connected neighbourhood of a pole q is naturally filtered

according to the order on the preimages q1, q2 of q.

Lemma 3.29

On any double intersection in SX that is incident to a pole q ∈ D, the canonical cocycle v(∇ab) is a

unipotent automorphism of the vector bundle π∗Eab with respect to the natural filtration of π∗Eab.

Definition 3.30 (Voros automorphism)

The automorphism vqb(∇ab) will be called a Voros automorphism. The unipotent matrix of a Voros

autormorphism vqb(∇ab) written in the decomposition π∗Eab ∼−→ Lq

1 ⊕ Lq2 will be called a Voros

matrix.

3. The detour path. We return to the calculation (3.15) in order to identify the only remaining

nontrivial map ∆ = I1h−11 : Lq

2 → Lq1 in the Voros automorphism (3.16). The automorphism

vqb(∇ab) is defined over the open subset ∆q

b of the triangular open set ∆ := ∆b. Let (p, q, r) be

the vertices of ∆. Let H ⊂ Σ be the hexagon above ∆; in symbols, π(H) = ∆. The vertices of

H are (p1, q2, r1, p2, q1, r2), as depicted in the diagram below on the left. In this diagram we also

indicate the representative L⋄i of Eab on the Levelt cell corresponding to each pole, as well as the

maps g1, g2, g3 which are some of the gluing maps for the abelianisation line bundle (Eab,∇ab) as

indicated. Notice that g1 = h−11 .


Using lemma 2.50 once again, we see that the map I1 : Lp1 → Lq

1 in (3.15) factors through Lr2,

so I1 = g3g2. Multiplying the matrices in (3.15), we find that the map ∆ : Lq2 → Lq

1 in the Voros

matrix (3.16) is given by

∆ = I1h−11 = g3g2g1 : Lq

2 → Lq1 . (3.17)

Since the maps g1, g2, g3 are just transition functions of the flat line bundle Eab, we can interpret the

isomorphism ∆ as the parallel transport of ∇ab along a clockwise semi-circular path δ(a) contained

in the punctured hexagon and connecting two preimages a2 to a1 of a point a on the base ∆qqb, as

indicated in the picture above on the right. Notice that the source a2 of δ(a) belongs to the Levelt

cell Hq2 and target a1 belongs to the Levelt cell Hq

1, as indicated in the picture above on the right.

Definition 3.31 (detour path, path-lifting rule)

For any point a ∈ ∆qb, let δqb(a) be the unique (homotopy class of a) path contained in the punctured

hexagon H× = H \b

going from a2 to a1 clockwise around b. The path δqb(a) will be called the

detour path. The map

δqb : ∆qb −→ Π1(H×) sending a 7−→ δqb(a)

will be called the path-lifting rule for ∆qb.


Proposition 3.32

The canonical cocycle v(∇ab) is fully determined by its values on the double intersections of the form

∆qb. For any such double intersection ∆q

b, the corresponding Voros matrix vqb(∇ab) can be expressed as

vqb(∇ab) =

[1 ∆

0 1

]:

Lq1 Lq

1

Lq2 Lq

2

⊕ ⊕∆ , where ∆ = Par(∇ab, δqb

), (3.18)

and where δqb : ∆qb → Π1(H×) is the path-lifting rule for ∆q

b.

3.4.3. The Voros Cocycle

In this subsection, we introduce the following shorthand notation:

Conn×X := Conn∗(X×,D, sl2; oD) ;

ConnX := Conn∗(X,D, sl2; oD) ; (3.19)

ConnΣ := Conn∗odd(Σ

×, π∗D, ab; 0abD ) .

Recall that the category Conn×X (resp. ConnX) consists of nice sl2-connections on (X×,D) (resp. on

(X,D)) with prescribed residue data oD along D, and Conn∗Σ consists of odd abelian connections on

(Σ×, π∗D) with prescribed residue data.

1. Universality of the canonical cocycle. The most important aspect of proposition 3.32 is the

fact that the expression (3.18) for vqb(∇ab) is a prescription for defining an automorphism that

depends only on the path-lifting rule δqb and not on the particular way that the connection ∇ab was

constructed. Therefore, we can apply this prescription to any abelian connection on Σ×, not just


the connection ∇ab obtained via the abelianisation functor. Indeed, given any abelian connection

(L,∇) ∈ Conn∗(Σ×, π∗D; ab), we can define a Cech 1-cocyle v(∇) ∈ Z1(SX,Aut(π∗L)

)by defining

an automorphism of (π∗L, π∗∇) over each double intersection of the form ∆qb by

vqb(∇) :=

[1 Par(∇, δqb)0 1

]:

L1 L1

L2 L2

⊕ ⊕ . (3.20)

Here, the preimage of the open set ∆qb on Σ is two disjoint open sets Hq

1|b,Hq

2|b, and Li is the

restriction of L to Hq

i|b.

2. The Voros cocycle. Consider the pushforward functor

π∗ : ConnΣ −→ Conn×X .

For every object (L,∇) ∈ ConnΣ, equation (3.20) determines an automorphism

(π∗L, π∗∇)vqb(∇)

−−−−→ (π∗L, π∗∇) . (3.21)

Furthermore, if φ : (L,∇) → (L′,∇′) is a morphism in Conn∗Σ, then the automorphisms vqb(∇), vqb(∇

′)

fit into the following commutative square over ∆qb:

(π∗L, π∗∇) (π∗L, π∗∇)

(π∗L′, π∗∇′) (π∗L′, π∗∇′)

π∗φ

vqb(∇)

π∗φ

vqb(∇′)

.

In other words, over each double overlap ∆qb, the family of automorphisms vqb :=

vqb(∇)

∣∣ ∇ ∈ ConnΣ

forms a natural automorphism of the pushforward functor π∗. Therefore, the collection of all such

natural automorphisms vqb, one for each double intersection in SX, forms a Cech 1-cocycle v with

values in the sheaf Aut(π∗) of groups of natural automorphisms of the pushforward functor π∗.


Definition 3.33 (Voros cocycle for SX)

The Cech 1-cocycle

v ∈ Z1(SX,Aut(π∗)

)defined by (3.20) is called the Voros cocycle for the Stokes open cover SX.

3. The deformed pushforward functor. The purpose of any cocycle v ∈ Z1(SX,Aut(π∗)

)is to

‘deform’ the pushforward functor π∗ in the following sense. Let (L,∇Σ) be an odd meromorphic

abelian connection on (Σ×, π∗D); that is, (L,∇Σ) ∈ ConnΣ. The pushforward functor

π∗ : ConnΣ −→ Conn×X

sends (L,∇Σ) to the connection (π∗L, π∗∇Σ) on (X×,D). Then the cocycle v ∈ Z1(SX,Aut(π∗)

)determines a cocycle

v(∇Σ) ∈ Z1(SX,Aut(π∗L)

).

Denote by (E ,∇X) the sl2-connection on (X×,D) obtained from (π∗L, π∗∇Σ) by applying the cocycle

v(∇Σ). The action of the cocycle v on the pushforward functor π∗ results in a new, deformed functor

πv∗ : ConnΣ −→ Conn×

X . (3.22)

which takes (L,∇Σ) to (E ,∇X).

4. The deabelianisation functor. Let v ∈ Z1(SX,Aut(π∗)

)be the Voros cocycle. Choose any odd

abelian connection (L,∇Σ) ∈ ConnΣ. Recall that the connection (π∗L, π∗∇) ∈ Conn×X has quasi-

permutation monodromy about the branch locus B and therefore does not admit a holomorphic

extension over B. Said differently, the image (π∗L, π∗∇) of (L,∇) under the usual pushforward

functor π∗ : ConnΣ → Conn×X does not lie in the subcategory ConnX ⊂ Conn×

X of connections on

(X,D). In contrast, we have the following proposition.


Proposition 3.34

The deformed pushforward functor πv∗ factors through the inclusion functor ConnX −→ Conn×X :

ConnΣ Conn×X

ConnX

πv∗

πvab

. (3.23)

Definition 3.35 (deabelianisation functor wrt the Voros cocycle)

The functor πvab : ConnΣ −→ ConnX in proposition 3.34 will be called the deabelianisation func-

tor with respect to the Voros cocycle v.

Proof of proposition 3.34.

Given (L,∇Σ) ∈ ConnΣ, denote by (E ,∇) its image under the functor πv∗ . To prove the proposition,

it is sufficient to show that the monodromy of ∇ about every branch point b ∈ B is trivial, because

any connection over X \ B with trivial monodromy about B admits a canonical extension to a

connection over X which is holomorphic along B.

1. The setup. Choose a branch point b ∈ B, and let ∆ := ∆b be the triangular open neighbourhood

of b. Let (p, q, r) be the vertices of ∆ arranged in this cyclic anti-clockwise order. This situation

is depicted below on the left. Consider the Stokes open cover of ∆ by three Stokes supersectors

∆pq,∆qr,∆rp, depicted below in the middle. Their double intersections are depicted below on the

right and denoted by:

∆p := ∆rp ∩ ∆pq , ∆q := ∆pq ∩ ∆qr , ∆r := ∆qr ∩ ∆rp .


Without loss of generality, fix a basepoint xp ∈ ∆ \ b and a loop γ ∈ π1(∆ \ b, xp) going anti-

clockwise around the branch point b, as pictured above on the left. Furthermore, subdivide the

loop γ accordingly: choose points xq ∈ γ ∩ ∆q and xr ∈ γ ∩ ∆r, and let γpq, γqr, γrp be the segments

of the path γ going from xp to xq and so on, as depicted above on the right. Introduce the following

notation for the corresponding parallel transports of π∗∇Σ:

Ppq := Parγpq(π∗∇Σ) , Pqr := Parγqr(π∗∇Σ) , Prp := Parγrp(π∗∇Σ) .

Let b ∈ R be the turning point on Σ above b, and let H := Hb ⊂ Σ be the hexagonal open

neighbourhood of b, and let (p1, q2, r1, p2, q1, r2) be the six vertices of H arranged in this cyclic

anti-clockwise order. The hexagon H is covered by six Stokes supersectors as usual. Denote the six

double intersections by Hpi ,H

qi ,Hr

i with i = 1, 2 such that π(H⋄i ) = ∆⋄ with ⋄ = p, q, r. This situation

is depicted in the following picture.

Denote the restriction of the line bundle L to H⋄i by L⋄

i . Denote by g1, . . . , g6 the isomorphisms

between the consecutive restrictions L⋄i obtained by parallel transport of ∇Σ; thus, for example,

g1 : Lp1

∼−→ Lq2 is the parallel transport of ∇0 along a path from Hp

1 to Hq2 that does not incircle the

turning point b. All this information is depicted in the following diagram.

g1

g2

g3

g4

g5

g6

Lp1

Lq2Lr

1

Lp2

Lq1 Lr

2


Since ∇Σ is an odd connection, its monodromy around the ramification point b is −1, so the

isomorphisms g1, . . . , g6 satisfy the following relationship:

g6 · · · g1 = −1 : Lp1

∼−→ Lp1 . (3.24)

2. Presentation of π∗L in this setup. With these definitions, we have the following canonical

presentations of π∗L over each double intersection ∆p,∆q,∆r:

π∗L∣∣∆p

=

Lp1

Lp2

⊕ , π∗L∣∣∆q

=

Lq2

Lq1

⊕ , π∗L∣∣∆r

=

Lr1

Lr2

⊕ ; (3.25)

In these presentations the parallel transports of π∗∇Σ along the paths γpq, γqr, γrp are:

Ppq =

[g1 0

0 g4

]:

Lp1 Lq

2

Lp2 Lq

1

⊕

g1

⊕

g4

,

Pqr =

[g2 0

0 g5

]:

Lq2 Lr

1

Lq1 Lr

2

⊕

g2

⊕

g5

, (3.26)

Prp =

[g3 0

0 g6

]:

Lr1 Lp

2

Lr2 Lp

1

⊕

g3

⊕

g6

.

Note that we expressed π∗L|∆q as Lq2 ⊕ Lq

1 rather than Lq1 ⊕ Lq

2 only in order to keep the three

matrices Ppq, Pqr, Prp in diagonal form.

Remark 3.36 (Monodromy of (π∗L, π∗∇Σ))

As a peripheral remark, let us compute the monodromy of (π∗L, π∗∇Σ) in the above presentation

of π∗L. Using the explicit expressions (3.26) and the relation (3.24), the monodromy of π∗∇0

along γ is readily seen to be a quasi-permutation monodromy, as expected:

Parγ(π∗∇Σ) = PrpPqrPpq =

[0 g3g2g1

− (g3g2g1)−1 0

]:

Lp1 Lp

1

Lp2 Lp

2

⊕ ⊕ .

This calculation reproduces a similar calculation found in [HN16, equation (4.5)].


3. The Voros matrices. We now compute the Voros matrices vp(∇Σ), vq(∇Σ), v

r(∇Σ). For each

open set ∆p,∆q,∆r, the path-lifting rule is well-defined, so we find:

V ⋄ := v⋄(∇Σ) =

[1 ∆⋄0 1

]:

L⋄1 L⋄

1

L⋄2 L⋄

2

⊕ ⊕∆⋄ , (3.27)

where the three isomorphisms ∆⋄ : L⋄2

∼−→ L⋄1 are defined by:

∆p := (g3g2g1)−1 , ∆q := (g1g6g5)

−1 , ∆r := (g5g4g3)−1 . (3.28)

These expressions can be easily computed by examining the following schematic pictures of the

path-lifting rule:

4. The monodromy of (E,∇). The cocycle v(∇Σ) acts on the fibre of π∗L above x⋄ by the


automorphism V ⋄, so the parallel transport of ∇ is computed as the following composition:

Parγ(∇) = Prp · V r · Pqr · V q · Ppq · V p :

Lp1 Lp

1 Lq2 Lq

2 Lr1 Lr

1 Lp2

Lp2 Lp

2 Lq1 Lq

1 Lr2 Lr

2 Lp1

⊕ ⊕

g1

⊕ ∆q ⊕

g2

⊕ ⊕

g3

⊕∆p

g4 g5

∆r

g6

.

Using the explicit expressions (3.26), (3.27), (3.28), as well as the relation (3.24), this composition

is easily computed:

Parγ(∇) =

[g3 0

0 g6

][1 ∆r

0 1

][g2 0

0 g5

][1 0

∆q 1

][g1 0

0 g4

][1 ∆p

0 1

]=

[0 1

1 0

];

in other words,

Parγ(∇) =

[1 0

0 1

]:

Lp1 Lp

1

Lp2 Lp

2

⊕ ⊕ .

Therefore, the monodromy of (E ,∇) about the branch locus B is trivial, and the proof of proposi-

tion 3.34 is complete.

3.5. Abelianisation Theorem

Let (X,D) be a logarithmic pair, fix generic residue data oD along D, and a nice saddle free quadratic

differential q ∈ H0X

(ω2X(2D)

)compatible with oD. Let π : Σ → X be the spectral cover determined

by q, and let oabD be the abelianisation residue data along π∗D.

3.5.1. Abelianisation with respect to the Stokes Open Cover SX

Theorem 3.37

The abelianisation functor πab is an equivalence of categories:


Moreover, there exists a canonical Cech 1-cocycle v ∈ Z1(SX,Aut(π∗)

)for the Stokes open cover SX

such that an inverse equivalence is given by the deabelianisation function πvab.

Proof.

We show directly that the functors πvab πab : ConnX → ConnX and πab πvab : ConnΣ → ConnΣ are


naturally isomorphic to the identity functors IdX, IdΣ on ConnX,ConnΣ, respectively.

Given (E ,∇) ∈ ConnX, consider (Eab,∇ab) = πab(E ,∇) ∈ ConnΣ. By definition, the deabelian-

isation functor πvab is obtained by applying the Voros cocycle v to the direct image functor π∗.

Specifically, the connection πvab(Eab,∇ab) ∈ ConnX is defined by applying the cocycle v(∇ab) to

Eab. So by proposition 3.28, πab(Eab,∇ab) = (E ,∇). So we define the component at (E ,∇) of the

natural isomorphism IdX ⇒ πvab πab to be the identity map id : (E ,∇) → (E ,∇). We must check

that this choice is natural. To do so, we must simply check that the composition πvabπabφ is the

identity on morphisms.

Let φ : (E ,∇) → (E ′,∇′) be a morphism in ConnX. Over every Stokes supersector ∆pqb ⊂ X, the

bundles E , E ′ are both equipped with a pair of transverse Levelt filtrations associated with p and q.

So we have natural decompositions E ∼−→ Lp1 ⊕ Lq

1 and E ′ ∼−→ L′p1 ⊕ L′q

1 . The map φ : E → E ′ is

Levelt filtration preserving by definition of ConnX, so with respect to the decompositions of E , E ′, it

is a diagonal map φ = diag(φ1, φ2) : Lp1 ⊕ Lq

1 → L′p1 ⊕ L′q

1 . Thus, upon identifying Lq1 and L′q

1 with

Lp2 = E

/Lq1 and L′p

2 = E/

L′q1 , we find that the induced map φ : gr(Ep

• ) → gr(E ′p• ) is also diagonal:

φ =

[φ1 0

0 φ2

]:

Lp1 L′p

1

Lp2 L′p

2

⊕

φ1

⊕

φ2

,

where φ2 is just φ2 precomposed by Lp2

∼−→ Lq1 and postcomposed by L′q

1∼−→ L′p

2 . The resulting

map πabφ : Eab → E ′ab is simply φ1 over the Levelt cell Up1 and φ2 over the Levelt cell Up

2. Therefore,

the pushforward π∗πabφ is again φ, which implies that πvabπabφ = φ.

Now, let (L,∇) ∈ ConnΣ, and consider its deabelinisation (Lab,∇ab) := πvab(L,∇). Over each

Levelt cell Up ⊂ X, the vector bundle Lab has a canonical presentation as π∗L which is naturally

a direct sum Lp1 ⊕ Lp

2 with Lpi := L|Up

i. This bundle is equipped with the Levelt filtration Lp

1 ⊂

(Lp1 ⊕ Lp

2). Then clearly the application of πab recovers L over Up1 and Up

2.

Remark 3.38

As evident from the proof of theorem 3.37, the pair πab, πvab is in fact an isomorphism of categories.


3.5.2. The Voros Class

1. Deabelianisation functors. Now we know that the abelianisation functor πab is an equivalence

of categories. As usual with categorical equivalences, an inverse equivalence to πab is not unique:

to construct one, the knowledge of πab alone is insufficient and additional choices need to be made.

Indeed, in theorem 3.37, the choice of a Cech cover SX was made in order to construct a Voros

cocycle v ∈ Z1(SX,Aut(π∗)

), which was used to build an inverse functor πvab. Therefore, there

may be many functors ConnΣ → ConnX which invert the abelianisation functor, at least up to

isomorphism. Each such inverse can be reasonably said to deabelianise a given abelian connection.

Definition 3.39 (deabelianisation functor)

Any functor ConnΣ → ConnX which is an inverse equivalence to the abelianisation functor πab is

called a deabelianisation functor.

The relationship between any two deabelianisation functors inverting πab is clear straight from the

definition of categorical equivalences.

Lemma 3.40

Any two deabelianisation functors for πab are naturally isomorphic.

2. Cech to sheaf cohomology. To eliminate the seeming dependence of the equivalence in the-

orem 3.37 on the chosen Stokes cover SX, we pass to sheaf cohomology. The Voros cocycle

v ∈ Z1(SX,Aut(π∗)

)determines a Cech cohomology class in H1

(SX,Aut(π∗)

). The Stokes open

cover SX is 1-acyclic for the sheaf Aut(π∗). Indeed, for every ∆pqb ∈ SX and any (L,∇) ∈ ConnΣ,

the vector bundle π∗L|∆pqb

is equipped with two transverse Levelt filtrations, π∗Lp, π∗Lq. Thus, any

1-cocycle ψ ∈ Z1(∆pqb ,Aut(π∗L)

)must be diagonal with respect to the decomposition π∗L coming

from the transverse intersection π∗Lp ⋔ π∗Lq of these filtrations. Then one can find a 0-cochain

that conjugates ψ to the identity id, so H1(∆pqb ,Aut(π∗L)

)= id. This is true for every (L,∇), so

H1(∆pqb ,Aut(π∗)

)= Id, where Id : π∗ ⇒ π∗ is identity natural automorphism of π∗.

Then using the analogue of Leray’s theorem for nonabelian cohomology, the Cech cohomology

set H1(SX,Aut(π∗)

)is equal to the sheaf cohomology set H1

(X×,Aut(π∗)

). Therefore, the Voros

cocycle v determines a sheaf cohomology class V := [v] ∈ H1(X×,Aut(π∗)

).


3. The Voros class. Suppose u ∈ V is another representative of the Voros class. It determines

another deformed functor

πu∗ : ConnΣ −→ Conn×X .

An obvious question is: what is the relationship between πv∗ and πu∗? Notice that a priori we do not

know if πu∗ factors through ConnX.

Proposition 3.41

Let V ∈ H1(X×,Aut(π∗)

)be any cohomology class, and choose a cocycle v ∈ V representing it.

Suppose the resulting deformed functor πv∗ : ConnΣ → ConnX is an inverse equivalence to the abelian-

isation functor πab : ConnX → ConnΣ. Then the deformed functor πu∗ for any other representative

u ∈ V is also an inverse equivalence to πab.

Definition 3.42 (Voros cocycle, Voros class)

Any cocycle v ∈ Z1(X×,Aut(π∗)

)with the property that the deformed functor πv∗ is an inverse

equivalence to π∗ will be called a Voros cocycle, and its cohomology class V = [v] ∈ H1(X×,Aut(π∗)

)will be called the Voros class.

4. Unipotent cocycles.

Definition 3.43 (open cover subordinate to (B,D))

A Cech open cover UX of X× is called subordinate to (B,D) if

• each open set in UX is incident to a single branch point in B and a pair of poles in D;

• each double intersection in UX is one of the following two types:

incident to a single branch point in B and a single pole in D;

incident to a pair of poles in D.

The Stokes open cover SX is subordinate to (B,D).

A Voros cocycle v ∈ Z1(X×,Aut(π∗)

)decomposed in any open cover subordinate to (B,D) is fully

determined by its values on open sets incident to a branch point b ∈ B and a single a pole p ∈

D. Given any flat line bundle (L,∇) ∈ ConnΣ, its pushforward π∗L over such an open set is

necessarily filtered, and the Voros cocycle v(∇) is an automorphism of this structure. As a result, a

Voros cocycle is necessarily a filtered automorphism (i.e., an upper-triangular matrix with respect


to the appropriate decomposition of π∗L). Moreover, the cohomological freedom provided by

proposition 3.41 allows us to modify v by a coboundary to bring it to a unipotent form, as we now

show.

Lemma 3.44

For any cover UX subordinate to (B,D), a Voros class V ∈ H1(X×,Aut(π∗)

)has a unipotent represen-

tative in Z1(UX,Aut(π∗)

).

Proof.

Choose any cocycle v ∈ V , and a Cech open cover UX of X× subordinate to (B,D). Choose two

intersecting open sets U,V ∈ U(1)

X , and let b ∈ B and p, q, r ∈ D be such that U is incident to b, p, q,

and V is incident to b, p, r. Denote by vpb ∈ Aut(π∗) the value of v on the double intersection U ∩ V.

Any Cech 0-cochain η ∈ C0(UX,Aut(π∗)

)determines a new cocycle u ∈ V by

upb = ηprb vpb(η

qpb )−1 , (3.29)

where ηprb is the value of η on the open set ∆prb , and likewise ηqpb .

Choose an abelian connection (L,∇) ∈ ConnΣ. Then (3.29) becomes an equality of automorphisms

of the flat vector bundle π∗L over the double intersection U ∩ V:

upb(∇) = ηprb (∇)vpb(∇)(ηqpb (∇)

)−1 .

The vector bundle π∗L has two transverse filtrations over U ∩ V, and the automorphism ηprb (∇)

must preserve both of them. Therefore, it must be diagonal in the decomposition π∗L = Lp1 ⊕ Lp

2.

Similarly, ηqpb (∇) must also be diagonal in this decomposition. Let:

ηprb (∇) =

[η1 0

0 η2

]:

Lp1 L′p

1

Lp2 L′p

2

⊕

η1

⊕

η2

and ηqpb (∇) =

[η3 0

0 η4

]:

Lp1 L′p

1

Lp2 L′p

2

⊕

η3

⊕

η4

.

At the same time, vpb(∇) is necessarily upper-triangular in this decomposition of π∗L, so let

vpb(∇) =

[a b

0 c

]:

Lp1 L′p

1

Lp2 L′p

2

⊕

a

⊕

c

b .


Then we find:

upb(∇) = ηprb (∇)vpb(∇)(ηqpb (∇)

)−1=

[η1 0

0 η2

][a b

0 c

][η−13 0

0 η−14

]=

[η1aη

−13 η1bη

−14

0 η2cη−14

].

So if we choose η3 := η1a and η4 := η2c, then upb(∇) becomes unipotent.

3.5.3. Abelianisation Theorem.

Let (X,D) be a logarithmic pair. Fix some generic sl2-residue data oD along D, as well as a nice

saddle-free quadratic differential q ∈ H1(X, ω2

X(2D)). Let π : Σ → X be the spectral cover deter-

mined by q. Consider the following categories:

ConnX := Conn∗(X,D, sl2; oD) ;

ConnΣ := Conn∗odd(Σ

×, π∗D, ab; 0abD ) .

Theorem 3.45 (Abelianisation Theorem)



Moreover, there exists a canonical cohomology class V ∈ H1(X×,Aut(π∗)

)such that any representa-

tive v ∈ V determines a functor πvab : ConnΣ −→ ConnX, called the deabelianisation functor, which is

an inverse equivalence to πab.

Proof.

The abelianisation functor πab was constructed in section 3.3.1 and lemma 3.24. Using the Stokes

open cover SX to compute the cohomology set H1(X×,Aut(π∗)

), in section 3.4.3 we construct a

canonical Cech 1-cocycle v ∈ Z1(SX,Aut(π∗)

)which determines a functor πv∗ : ConnΣ → ConnX.

By theorem 3.37, πv∗ is an inverse equivalence to πab, and by proposition 3.41 any other represen-

tative u of the cohomology class V := [v] ∈ H1(X×,Aut(π∗)

)determines an inverse equivalence to

πab.

Let MX,MΣ be the moduli spaces corresponding to ConnX,ConnΣ. As an equivalence of categories,

the abelianisation functor πab preserves equivalence classes, and therefore readily descends to a


bijection at the level of moduli spaces. Moreover, although there may be many deabelianisation

functors ConnΣ → ConnX, they all descend to the same bijection at the level of moduli spaces,

because they are inverse equivalences to the same functor πab.

Corollary 3.46 (Abelianisation Correspondence)

There exists a bijection of moduli spaces πab : MX∼−→ MΣ, called the abelianisation map, given

explicitly by the abelianisation functor πab. Its inverse, πab : MΣ∼−→ MX, called the deabelianisation

map, can be given explicitly by choosing a Voros cocycle v ∈ Z1(X×,Aut(π∗)

)and only depends on

the cohomology class [v] ∈ H1(X×,Aut(π∗)

).

chapter 4

Quantum Differential Operators

Contents4.1 The Quantum Family . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93

The ℏ-space. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93

The quantum family. . . . . . . . . . . . . . . . . . . . . . . . . . . . 93

The quantum tangent sheaf. . . . . . . . . . . . . . . . . . . . . . . . 94

4.2 Quantum Connections and Quantum Differential Operators . . . . . . . . . . . 94

4.2.1 Quantum Connections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95

Local expression. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95

The category of quantum connections. . . . . . . . . . . . . . . . . . 95

Quantum residue data. . . . . . . . . . . . . . . . . . . . . . . . . . . 96

Quantum connections with fixed residue data. . . . . . . . . . . . . . 98

Quantum Jets. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98

4.2.2 Classical Spectral Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98

4.3 Quantum Differential Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99

Germs of quantum connections. . . . . . . . . . . . . . . . . . . . . . 99

Quantum differential systems. . . . . . . . . . . . . . . . . . . . . . . 100

4.3.1 Local Normal Form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100

(1) Spectral Splitting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102

Constructing a formal solution. . . . . . . . . . . . . . . . . . . . . . 103

Convergence of the formal solution. . . . . . . . . . . . . . . . . . . . 104

Conclusion. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104

(2) Quantum Spectral Splitting . . . . . . . . . . . . . . . . . . . . . . . . . 104

Constructing an ℏ-formal solution. . . . . . . . . . . . . . . . . . . . 105

G is a uniformly Gevrey formal power series. . . . . . . . . . . . . . 106

(3) Asymptotic Existence . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106

(4) Removing the Smooth Part . . . . . . . . . . . . . . . . . . . . . . . . . 107

4.4 Quantum Levelt Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107

Generic quantum residue data. . . . . . . . . . . . . . . . . . . . . . 107

Existence of quantum Levelt filtration. . . . . . . . . . . . . . . . . . 108

4.4.1 Nice Quantum Connections . . . . . . . . . . . . . . . . . . . . . . . . . . . 111

4.5 The WKB Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112

4.5.1 The WKB Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112

The WKB problem. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112

The WKB method. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112

The WKB splitting. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114

4.5.2 The Schrodinger Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115

Quantum spin structure. . . . . . . . . . . . . . . . . . . . . . . . . . 115

91

TOC | chapter 4 | 92

Quantum Anti-Spin Structure and Its Quantum Jets. . . . . . . . . . 116

Connections on Quantum Jets. . . . . . . . . . . . . . . . . . . . . . 117

The Schrodinger Connection. . . . . . . . . . . . . . . . . . . . . . . 117

The WKB method. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118

4.5.3 Opers and Transverse Extensions . . . . . . . . . . . . . . . . . . . . . . . . 119

The jet filtration. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120


4.1. The Quantum Family

1. The ℏ-space. Let ℏ be the linear coordinate on the complex affine line A1 with analytic topol-

ogy; we will habitually denote this space by A1ℏ . Let S be a germ of a real-oriented sectorial

neighbourhood at 0 ∈ A1ℏ with opening A.

This means we have the following data. We consider the real-oriented blowup [A1ℏ : 0] of A1

ℏ at

the origin 0, and we fix an arc A in the blowup circle S0(A1ℏ), which in the case of the affine line is

canonically identified with the standard circle S1. Then any real-oriented sectorial neighbourhood

S representing the germ S is an open subset S ⊂ [A1ℏ : 0] which by definition contains the arc A

itself. Denote by S× := S \ A ⊂ A1ℏ \ 0 the punctured sectorial neighbourhood, and by S× the

corresponding germ.

The germ S is quipped with the ring GSℏ of germs of Gevrey functions on S. Recall that these are

germs of holomorphic functions on the sector S× with the property that they admit an asymptotic

expansion as ℏ → 0 of Gevrey class.

2. The quantum family. Let C be a smooth complex irreducible projective curve, and consider

the trivial familyX := C× S

S .

pr

We will often refer to this data as a quantum family. For any point ℏ ∈ S, we denote1 the fibre of pr

above ℏ by X(ℏ); so X(ℏ) := pr−1(ℏ). It is a copy of C. We will sometimes refer to the fibre X(0) as the

classical curve. On the whole, our terminology will employ the adjectives quantum and classical to

refer respectively to “things spread over the ℏ-space” and “things sitting over ℏ = 0”. We will also be

interested in sections of the quantum family X → S. Thus, to be consistent with our terminology, the

image p ⊂ X of a section p : S → X will be referred to as a quantum point: it is a family of points

depending on ℏ. Its intersection with a fibre X(ℏ) will be denoted by p(ℏ); so p(ℏ) := p ∩ X(ℏ) = p(ℏ).

In particular, its intersection with the classical curve, p(0) := p ∩ X(0) = p(0), will be referred to as

a classical point2. A quantum divisor D ⊂ X is a finite linear combination of disjoint quantum

points with integer coefficients. We will say that a quantum divisor D is horizontal if D = D(0) × S

1The notation “X(ℏ)” is inspired by the standard notation in perturbation theory.2Thus, a quantum point is actually a 1-dimensional object, whilst a classical point is a 0-dimensional object.


for some divisor D(0) ⊂ X(0).

Because X has a simple product structure, the ring of functions on X is simply OX = OC⊗OS. Thus,

if x is a local coordinate on C, a function f(x, ℏ) is an element of OX if it is holomorphic in x and

of Gevrey class in ℏ. For brevity, we will refer to sections of OX as quantum functions. We will say

that a function f ∈ OX is vertically constant if in fact f ∈ OS, and it is horizontally constant if in

fact f ∈ OC.

3. The quantum tangent sheaf. Consider the corresponding relative tangent sequence of OX-

modules:

0 TX/S TX ⟨∂ℏ⟩ 0 ,pr∗ (4.1)

where ∂ℏ is the canonical generator of the tangent sheaf TA1ℏ

restricted to S. The central object

of this thesis — whose representations will be studied extensively in the subsequent sections and

chapters — is the twist of the relative tangent sheaf TX/S by the classical curve X(0).

Definition 4.1 (quantum tangent/cotangent sheaf)

The quantum tangent sheaf on the family X is the OX-module

T ℏX/S := TX/S

(− X(0)

)consisting of relative vector fields vanishing at ℏ = 0. The quantum cotangent sheaf is its dual

Ω1,ℏX/S

:=(T ℏX/S

)∨= Ω1

X/S

(X(0))

,

consisting of relative 1-forms with a simple pole at ℏ = 0.

The quantum tangent sheaf T ℏX/S is a line bundle: if ∂x is a local generator of TC, then T ℏ

X/S is

locally generated by ℏ∂x. In fact, T ℏX/S has a natural map to the relative tangent sheaf TX/S which

is generically an isomorphism but drops rank along the divisor X(0) ⊂ X. Similarly, Ω1,ℏX/S is a line

bundle: if dx is a local generator for Ω1X/S, then Ω1,ℏ

X/S is locally generated by dx /ℏ.

4.2. Quantum Connections and Quantum Differ-

ential Operators


4.2.1. Quantum Connections

Let X → S be a quantum family, and fix a quantum divisor D ⊂ X. We introduce the following

shorthand notation for the quantum tangent and cotangent sheaves twisted by OX(logD):

AX/S := T ℏX/S(− logD) and/so A∨

X/S := Ω1,ℏX/S(logD) .

Definition 4.2 (quantum connection)

If E is a vector bundle on X, and D ⊂ X is a quantum divisor, a meromorphic quantum connection

on E with poles along D is an OS-linear map

∇ : E −→ A∨X/S ⊗

OX

E ,

which satisfies the quantum Leibniz rule

∇(fe) = ℏ df ⊗ e+ f∇e for all f ∈ OX, e ∈ E ,

where d : OX → Ω1X/S is the relative de Rham differential.

1. Local expression. To be concrete, let us choose a local coordinate chart (U;x, ℏ) near a quan-

tum point p ⊂ D, such that p is given by x = 0 × S. Then in terms of a local trivialisation of E

over U, a quantum connection ∇ with order r pole at p has the following form:

∇ =loc d−A(x, ℏ)dx

xrℏ, (4.2)

where A(x, ℏ) is an n× n matrix of quantum functions on U, and the relative de Rham differential

d is differentiation with respect to x only. We will establish some local normal forms in section 4.3.

2. The category of quantum connections. A morphism of two quantum connections φ : (E ,∇) →

(E ′,∇′) on (X,D) is an OX-linear map intertwining the quantum connections ∇,∇′, just as in

(2.1). The category of quantum connections on (X,D) will be denoted by QConn(X,D). The

subcategory of quantum connections on (X,D) on vector bundles of rank n will be denoted by

QConn(X,D; gln).


Let (E ,∇) ∈ QConn(X/S,D). By fixing some nonzero value of ℏ, the restriction of (E ,∇) to the

fibre X(ℏ) of X over ℏ ∈ S× is a usual connection on the curve X(ℏ) with poles bounded by the

divisor D(ℏ) = X(ℏ) ∩ D (this can be see from the explicit local expression equation 4.2, for exam-

ple). We shall denote the restriction of (E ,∇) to the fibre X(ℏ) by (E(ℏ),∇(ℏ)). Thus, any quantum

connection (E ,∇) is an ℏ-family of usual connections (E (ℏ),∇(ℏ)) ∈ Conn(X(ℏ),D(ℏ)). However, since

quantum connections are defined only over a germ of a sector S, a functor QConn(X/S,D) −→

Conn(X(ℏ),D(ℏ)) does not exist.

At the same time, as will be discussed in section 4.2.2, the restriction of any (E ,∇) ∈ QConn(X/S,D)

to the classical fibre X(0) = C above ℏ = 0 is a meromorphic Higgs bundle (E (0), ϕ) on C with poles

bounded by D(0). So we have one more functor

QConn(X/S,D) −→ Higgs(C,D(0))

(E ,∇) 7−→ (E (0), ϕ) ,

and this functor is also essentially surjective, and induces a surjective map

QMdR(X/S,D) −→ MHiggs(C,D(0)) .

Thus, QConn(X/S,D) is a category over S whose fibre above ℏ = 0 is the category Conn(X(ℏ),D(ℏ)),

and whose fibre over ℏ = 0 is the category Higgs(C,D(0)). Correspondingly, QMdR(X/S,D) is a

moduli space over S whose fibre above ℏ = 0 is the moduli space MdR(X(ℏ),D(ℏ)), and whose fibre

above ℏ = 0 is the moduli space MHiggs(C,D(0)). Similar considerations were first made by Simpson

in [Sim94, page 87, τ -connections] as suggested to him by Deligne.

3. Quantum residue data. Let (E ,∇) be a quantum connection on (X,D). Then ∇ has a well-

defined restriction

∇∣∣D: E∣∣D−→ E

∣∣D⊗OD

A∨X/S

∣∣D

,

which is OD-linear; i.e., it defines a section ∇|D ∈ End(

E |D)⊗ A∨

X/S

∣∣∣D

, which we call the quantum

polar Higgs field. If D is reduced (so ∇ is a logarithmic quantum connection), then we have a


short exact sequence analogous to the residue sequence (2.7):

0 Ω1,ℏX/S Ω1,ℏ

X/S(D) OD 0

m⊕i=1

OSdef=

m⊕i=1

GSℏ ,

Res

= (4.3)

where m is the number of quantum points (i.e,. the number of irreducible components of D). For

every quantum point p ⊂ D, we have

A∨X/S

∣∣p= Ω1,ℏ

X/S(D)∣∣p

∼−→ GSℏ ,

where the isomorphism is given by the residue map in (4.3). Moreover, the restriction E |p of the

bundle E to p is a GSℏ-module, so

End(

E |p)⊗ A∨

X/S

∣∣p

∼−→ End(

E |p)

.

Thus, the restriction ∇|p is an endomorphism of E |p, which we may call the quantum residue of

∇ at p, in analogy with the classical case.

Upon choosing a trivialisation of E |p, we get an isomorphism

End(

E |p) ∼= gln(ℏ)

def= gln ⊗CGSℏ ,

so the restriction ∇|p determines an element of gln(ℏ). Changing the trivialisation of E |p amounts

to conjugating this element by a gauge transformation of E |p which is Gevrey in ℏ by definition.

This means that for every fixed value of ℏ, the restriction ∇|(ℏ)p determines an adjoint orbit o(ℏ)p in

gln just as in the classical case, and these orbits vary in ℏ in a controlled way in the following sense.

In a trivialisation of E |p, for every ℏ ∈ S, the restriction ∇|(ℏ)p determines an element κ(ℏ) ∈ o(ℏ)p ; as

a matrix function of ℏ, the element κ(ℏ) is Gevrey in S.

Definition 4.3 (quantum residue data)

For every quantum point p ⊂ D, choose an ℏ-family of adjoint orbits op in gln(ℏ). The collection of

all these adjoint orbits,

oD :=op∣∣ p ∈ D

,

will be called quantum residue data.


4. Quantum connections with fixed residue data. Just as in the case of usual connections, the

quantum residue data is an invariant of a quantum connection.

Definition 4.4 (category of quantum connections with fixed quantum residue data)

We denote by QConn(X/S,D; gln, oD) the full subcategory of QConn(X,D; gln) consisting of quan-

tum gln-connections on (X,D) with fixed residue data oD. We denote the corresponding moduli

space by QMdR(X/S,D; gln, oD).

In other words, if (E ,∇) ∈ QConn(X,D; gln, oD), then for every point p(ℏ) ∈ D(ℏ), the restriction

∇|(ℏ)p lies in the adjoint orbit o(ℏ)p ∈ o(ℏ)

D .

5. Quantum Jets. Let DℏX/H denote the universal enveloping OX-algebra of the quantum tangent

sheaf T ℏX/H. Sections of Dℏ

X/H are called quantum differential operators. Like the usual sheaf

of differential operators, quantum differential operators are naturally filtered by order, Dℏ,⩽•X/H →

Dℏ,⩽•+1X/H .

Definition 4.5 (quantum jets)

If E is an OX-module, we define the sheaf of quantum k-jets of E to be the OX-module

J kℏ E := HomOX

(Dℏ,⩽k

X/H , E)

.

4.2.2. Classical Spectral Data

Let (E ,∇) be a quantum connection on (X,D). The restriction of ∇ to the clasical fibre C = X(0)

over ℏ = 0 is an OC-linear map

∇(0) : E (0) −→ E (0) ⊗OC

Ω1C(D

(0)) .

In other words, (E (0),∇(0)) is a meromorphic Higgs bundle on the curve C with poles along D(0).

Let a(0) := (a(0)

1 , . . . , a(0)n ) be the spectral data of ∇(0). The data a(0) is an invariant of quantum

connections.


Definition 4.6 (classical spectral data)

If (X,D) is a quantum pair, an n-tuple of sections

a(0) := (a(0)

1 , . . . , a(0)n ) where a(0)

k ∈ H0C

(ωC(k · D(0))

)will be called classical spectral data. The curve Σ(0) determined by the classical spectral data will

be called the classical spectral curve; its ramification locus will be referred to as the classical

turning points.

Definition 4.7 (category of quantum connections with fixed quantum residue data and clas-

sical spectral data)

We denote by QConn(X/S,D; gln, oD, a(0)) the full subcategory of QConn(X,D; gln, oD) consisting

of quantum gln-connections on (X,D) with fixed residue data oD and fixed classical spectral data

a(0). We denote the corresponding moduli space by QMdR(X/S,D; gln, oD, a(0)).

4.3. Quantum Differential Systems

1. Germs of quantum connections. We will often consider germs of quantum connections at

a quantum point. To be precise, if p ⊂ X is a quantum point, the ring of germs of quantum

functions at p is the sheaf3

OX,p :=⊔ℏ∈S

OX,p(ℏ) .

Correspondingly, introduce the following notation:

Ω1p := OX,p ⊗

OX

Ω1X/S and Ω1,ℏ

p := OX,p ⊗OX

Ω1,ℏX/S .

A germ of a meromorphic quantum connection at p is thus a free OX,p-module Ep of finite rank,

together with an OS-linear morphism ∇ : Ep → Ω1,ℏp (r · p) ⊗ Ep (for some nonnegative integer r)

satisfying the quantum Leibniz rule.

3Recall that p(ℏ) def= p ∩ X(ℏ) is a point.


2. Quantum differential systems. In a local chart (x, ℏ) such that p = x = 0 × S, we have the


OX,p∼= CSx, ℏ , Ω1

p∼= CSx, ℏdx , Ω1

p(r · p) ∼= CSx, ℏdx

xr.

Ω1,ℏp

∼= CSx, ℏdx

ℏ, Ω1,ℏ

p (r · p) ∼= CSx, ℏdx

xrℏ.

Recall that CSx, ℏ def= Cx⊗C GSℏ. A free OX,p-module Ep of rank n is thus a free module over

CSx, ℏ of rank n. A trivialisation of Ep is a choice of an isomorphism Ep∼−→ CSx, ℏ ⊗ Cn.

A quantum connection (E ,∇) obviously gives rise to a quantum connection germ at any quantum

point p ⊂ X, with Ep being OX,p ⊗OXE; i.e., the ring of germs of sections of E at the quantum point

p. According to our nomenclature, a trivial vector bundle is a vector bundle E which is isomorphic

to OnX, but the trivial vector bundle means that such a trivialisation E ∼−→ On

X has been specified.

Definition 4.8 (quantum differential system)

A germ of a meromorphic quantum differential system — or a quantum differential system

for short — of rank n is a germ of a meromorphic quantum connection at a quantum point p on

the trivial vector bundle.

The set of all meromorphic quantum systems at p is isomorphic to End(Cn)⊗ Cx[x−1]⊗GSℏ.

Definition 4.9 (local quantum gauge transformations)

We introduce the following gauge groups.

• The group of local quantum gauge transformations is Gℏ := GLn(Cx

)⊗GSℏ.

• The group of local ℏ-formal gauge transformations is GJℏK := GLn(Cx

)⊗GJℏK.

• The group of local x-formal gauge transformations is Gℏ := GLn(CJxK)⊗GSℏ.

• The group of local formal gauge transformations is GJℏK := GLn(CJxK)⊗GJℏK.

4.3.1. Local Normal Form

We study logarithmic quantum differential systems, which are objects of the form

α := A(x, ℏ)dx

xℏ,


where A(x, ℏ) is an n × n matrix of quantum functions; i.e., the entries of A(x, ℏ) are elements

aij(x, ℏ) ∈ Cℏ ⊗ GSℏ of the ring of germs of functions which are holomorphic at x = 0 and

Gevrey in the sector S. We will be concerned with real parts of the eigenvalues of the quantum

residue of α, which is the matrix A(0, ℏ)/ℏ of Gevrey germs.

Proposition 4.10 (Local Normal Form)

Let α be a logarithmic quantum differential system

α := A(x, ℏ)dx

xℏ,

where A satisfies the following properties:

(1) A(0, 0) is diagonalisable with distinct eigenvalues;

(2) the eigenvalues of A(0, 0)e−iϑ have distinct real parts for all phases ϑ ∈ arc(S);

(3) the sector S is sufficiently narrow that there exists a positive real number µ > 0 such that∣∣Re ((λi − λj)e−iϑ)∣∣ ⩾ µ for every pair of distinct eigenvalues λi, λj of A(0, 0) and every phase

ϑ ∈ arc(S).

Then α is equivalent via a local quantum gauge transformation G ∈ Gℏ to a diagonal quantum system

of the formα := Λ(x)

dx

xℏ+K(ℏ)

dx

x,

where Λ(0) = diag(λ1, . . . , λn) is the diagonal matrix of eigenvalues of A(0, 0) which are ordered by

their real parts according to Re(λ1e

−iϑ0)< · · · < Re

(λne

−iϑ0), where ϑ0 is the bisector of S.

Of course, assumption (3) implies (2), but we stated them separately for clarity. Notice that as-

sumption (2) implies that the sector S is sufficiently narrow that the order relation on the eigen-

values, given by Re(λ1e

−iϑ0)< · · · < Re

(λne

−iϑ0), is maintained throughout the sector S. These

assumptions will play a key role in the proof.

The proofs proceeds in four main steps:

(1) Spectral splitting: Diagonalise the ℏ-leading order term A(x, 0) by constructing an x-formal

gauge transformation G(0); then we show G(0) is actually holomorphic at x = 0.

(2) Quantum spectral splitting: Using the fact that A(x, 0) is diagonal, we diagonalise the re-

maining part A(x, ℏ) by constructing an ℏ-formal gauge transformation G; then we show that


G is a uniformly Gevrey formal series.

(3) Asymptotic existence: We use a combination of asymptotic existence theorems found in

[Was76] to show that there exists a quantum gauge transformation G which diagonalises

α and whose asymptotic expansion is precisely G.

(4) Removing the smooth part: Finally, we remove the smooth part of the resulting diagonal

connection by straightforward integration.

(1) Spectral Splitting

We begin by examining the ℏ-leading order part of α:

α(0) := A(x, 0)dx

x.

We diagonalise the matrixA(0) := A(x, 0) via an algorithm sometimes referred to as spectral splitting

[Boa99, Appendix B]. Without loss of generality, we assume that A(0)

0 := A(0, 0) is already diagonal

with eigenvalues ordered as in the hypothesis, which can be achieved by means of a constant gauge

transformation.

Next, we split A(0)

0 into two diagonal blocks, and write A(0) accordingly in block-matrix form:

A(0)

0 =:

A110 0

0 A220

and A(0)(x) =:

A11(x) A12(x)

A21(x) A22(x)

, (4.4)

for some holomorphic functions A11(x), A12(x), A21(x), A22(x).

We seek a gauge transformation G(0) = G(0)(x) which puts α(0) in diagonal form

α

(0)

:= Λ(0)dx

xrwhere Λ(0)(x) :=

Λ11(x) 0

0 Λ22(x)

, (4.5)

for some holomorphic matrices Λ11(x),Λ22(x). We will search for G(0) in the form

G(0)(x) =:

1 G12(x)

G21(x) 1

. (4.6)


If G(0) is to gauge transform α(0) intoα(0), G(0) and Λ(0) must satisfy the algebraic equation

A(0)G(0) −G(0)Λ(0) = 0 . (4.7)

Writing A(0),Λ(0), G(0) in their block-matrix forms (expressions (4.4 – 4.6)), equation (4.7) results

in the following four equations:

Λ11 = A11 +A12G21 ; A12 +A11G12 −G12A22 −G12A21G12 = 0 ;

Λ22 = A22 +A21G12 ; A21 +A22G21 −G21A11 −G21A12G21 = 0 .(4.8)

We focus our attention on solving the top right equation, which is the equation for G12. Also, to

simplify notation, we drop the superscript “12” in “G12” for the time being; thus, we wish to solve

for G the following algebraic equation:

A12 +A11G−GA22 −GA21G = 0 . (4.9)

1. Constructing a formal solution. We begin by writing each Aij(x) as a power series in x, and

solving equation 4.9 in formal power series:

Aij(x) =

∞∑k=0

Aijk x

k and G(x) :=

∞∑k=0

Gkxk . (4.10)

Substituting these expansions into equation 4.9, and separating order by order in x, we find:

x0 | A120 +A11

0 G0 −G0A220 −G0A

210 G0 = 0

where the faded terms are 0, because A120 = 0, A21

0 = 0. Thus, we can take G0 := 0. Next:

x1 | A121 +A11

1 G0 +A110 G1 −G1A

220 −A11

1 G0 −G1A210 G0 −G0A

211 G0 −G0A

210 G1 = 0 ,

i.e., A110 G1 −G1A

220 = −A12

1 , (4.11)

where again the faded terms are all 0, because G0 = 0. Since A110 , A

220 have by assumption no

eigenvalues in common, there exists a unique matrixG1 that satisfies this equation. More generally:

xk | A110 Gk −GkA

220 = −A12

k −k−1∑s=1

(A11

k−sGs −Gk−sA22s

)+

k−1∑s=1

k−s−1∑m=1

GmA21k−m−sGs . (4.12)


Again, by assumption, A110 , A

220 have no eigenvalues in common, so there exists a unique matrix Gk

that satisfies this equation. This completes the construction of G.

2. Convergence of the formal solution. In fact, the power series G has a positive radius of

convergence at x = 0. This follows from the fact that A(0)(x) is holomorphic at x = 0, so the

coefficients Aijk in the power series for Aij(x) have are bounded in an appropriate way. The lefthand

side of (4.12) is linear in the entries of the matrix Gk, so the lefthand side can be written in the

form A0Gk, where A0 is a nonsingular matrix whose entries are linear combinations of the entries

of A110 , A

220 ; and Gk is a column vector whose entries are the entries of Gk (in some unspecified

order). Thus, if Ak is a column vector whose entries are the entries of the righthand side of (4.12)

(in the same order as for Gk), then Gk = A−10 Ak. This observation allows us to deduce a bound

on the coefficients Gk of G from the bounds on Aijk . We refer the reader to section A.1 for more

details.

3. Conclusion. Returning to the notation of equations (4.8), we conclude that G12 — and hence

Λ22 — is holomorphic at x = 0. The exact same argument shows that G21, and hence Λ11, is

holomorphic at x = 0. Therefore, we conclude there exists a holomorphic gauge transformation

G(0) of the form (4.6) satisfying equation (4.7).

(2) Quantum Spectral Splitting

Assume that the ℏ-leading order part A(0)(x) of α has already been completely diagonalised using

for example the spectral splitting algorithm described above. We will diagonalise the remaining

part of the quantum differential system α using a very similar algorithm, but now performed with

respect to the ℏ-expansion of α. By analogy, we call it the quantum spectral splitting.

Split A(0)(x) into two diagonal blocks, and write A(x, ℏ) accordingly in block-matrix form:

A(0)(x) =:

A(0)

11 (x) 0

0 A(0)

22 (x)

and A(x, ℏ) =:

A11(x, ℏ) A12(x, ℏ)

A21(x, ℏ) A22(x, ℏ)

. (4.13)

We seek a quantum gauge transformation G = G(x, ℏ) which transforms α into

α := d− Λ(x, ℏ)dx

xrℏwhere Λ(x, ℏ) :=

Λ11(x, ℏ) 0

0 Λ22(x, ℏ)

, (4.14)


for some quantum matrices Λ11,Λ22 to be determined in the course of the construction. We will

search for G in the form

G(x, ℏ) =

1 G12(x, ℏ)

G21(x, ℏ) 1

. (4.15)

If G is to gauge transform α into α, G and Λ must satisfy the differential equation

xℏ∂xG = AG−GΛ . (4.16)

Writing A,Λ, G in their block-matrix forms (expressions (4.13 - 4.15)), equation (4.16) results in

the following four equations4:

Λ11 = A11 +A12G21 ; xℏ∂xG12 = A12 +A11G12 −G12A22 −G12A21G12 ;

Λ22 = A22 +A21G12 ; xℏ∂xG21 = A21 +A22G21 −G21A11 −G21A12G21 .

We focus our attention on solving the top right equation, which is a differential equation for G12.

To simplify notation, we drop the subscript “12” in “G12” for the time being; thus, we wish to solve

for G the following equation:

xℏ∂xG = A12 +A11G−GA22 −GA21G . (4.17)

1. Constructing an ℏ-formal solution. First, we will solve equation (4.17) in ℏ-formal power

series:

G(x, ℏ) :=∞∑k=0

G(k)ℏk . (4.18)

Each Aij(x, ℏ) is by assumption Gevrey in the sector S, so it has an asymptotic expansion:

Aij ∼ Aij :=

∞∑k=0

A(k)

ij ℏk as ℏ → 0 in S. (4.19)

We plug these expansions into equation (4.17), and separate order by order in ℏ to find:

ℏ0 | 0 = A(0)

12 + A(0)

11G(0) −G(0)A(0)

22 −G(0)A(0)

21G(0)

4The two equations in the second line are identical to the two equations in the first line with only all indices exchangedas 1 ↔ 2.


where the faded terms are all 0, because A(0)

12 = 0, A(0)

21 = 0. Thus, we can take G(0) := 0. Next:

ℏ1 | x∂xG(0) = A(1)

12 +A(1)

11G(0) +A(0)

11G(1) −G(1)A(0)

11 −G(0)A(1)

11

−G(1)A(0)

21G(0) −G(0)A(1)

21G(0) −G(0)A(0)

21G(1) ;

i.e., A(0)

11G(1) −G(1)A(0)

11 = −A(1)

12 , (4.20)

where again the faded terms are all 0, because G(0) = 0. Since A(0)

11 (x), A(0)

22 (x) are holomorphic at

x = 0, and A(0)

11 (0), A(0)

22 (0) have by assumption no eigenvalues in common, there exists a neighbour-

hood U of x = 0, such that A(0)

11 (x), A(0)

22 (x) remain having no eigenvalues in common for all x ∈ U.

Equation (4.20) is linear in the entries of G(1), so there exists a unique matrix G(1)(x), holomorphic

in U, that satisfies this equation. Importantly, U depends only on the ℏ-leading order part of A(x, ℏ).

More generally:

ℏk | A(0)

11G(k) −G(k)A(0)

22 = −A(k)

12 −k−1∑s=1

(A(k − s)

11 G(s) −G(k − s)A(s)

22

)+

k−1∑s=1

k−s−1∑m=1

G(m)Ak−m−s21 G(s) + x∂xG

(k − 1) . (4.21)

Once again, the lefthand side of equation (4.21) is linear in the entries of G(k), so there exists a

unique matrix G(k)(x), holomorphic in U, which satisfies this equation. This completes the con-

struction G(x, ℏ).

2. G is a uniformly Gevrey formal power series. By assumption, A(x, ℏ) is uniformly Gevrey in

the sector S, which means that the expansion coefficients A(k)

ij in (4.19) are appropriately bounded,

and these bounds are uniform for all x ∈ U. Using an argument very similar to the one presented

in paragraph 2, similar bounds can be deduced for the coefficients G(k) of G, which are likewise

uniform for all x ∈ U. It follows then that G is a power series in ℏ which is uniformly Gevrey for

x ∈ U. We refer the reader to section A.2 for more details.

(3) Asymptotic Existence

We now argue that there exists a quantum gauge transformation G(x, ℏ) which transforms the

quantum differential system α into a the diagonal system

α := Λ(x, ℏ)dx

xℏ, (4.22)


where Λ(x, ℏ) is a diagonal quantum matrix, such that G(x, ℏ) has the property that its asymptotic

expansion as ℏ → 0 in S is precisely the ℏ-formal gauge transformation G constructed previously.

The proof of this fact consists of combining ideas from the proofs of the Main Asymptotic Existence

Theorem [Was76, Theorem 12.1] and another asymptotic existence theorem [Was76, Theorem

26.1]. One key point to highlight is that the somewhat ad-hoc hypothesis (3) in our proposition

(proposition 4.10) ensures that the conclusion of Theorem 26.1 in [Was76] holds in the same

sector S in A1ℏ as in the hypothesis. Finally, since G is a uniformly Gevrey formal power series in S,

it follows that G(x, ℏ) behaves in ℏ necessarily as a matrix of germs of Gevrey functions in S.

(4) Removing the Smooth Part

The last step is to separate out the smooth part of the quantum differential system α in (4.22):

α := Λ(x)dx

xℏ+K(ℏ)

dx

x+B(x, ℏ) dx ,

where K(ℏ), B(x, ℏ) is are diagonal quantum matrices. Integrating B(x, ℏ) dx and exponentiating

the result, we obtain a quantum gauge transformation

˜G(x, ℏ) := exp

x∫0

B(x, ℏ) dx

.

Applying ˜G to α a new quantum differential systems:

α := Λ(x)

dx

xℏ+K(ℏ)

dx

x..

In conclusion, the quantum gauge transformation

G := ˜G G

transforms α into its local normal formα, which completes the proof of proposition 4.10.

4.4. Quantum Levelt Data

1. Generic quantum residue data. Choose a sector S ⊂ A1ℏ. Let g be either gln or sln, and

let g(ℏ) := g ⊗ GSℏ. Fix a germ of a regular semisimple element Λ(ℏ) ∈ g(ℏ), which means


that Λ(ℏ) is an n × n matrix of germs of Gevrey functions in S such that the leading term Λ(0)

is diagonalisable with distinct eigenvalues. We will say that Λ(ℏ) is generic if the eigenvalues

of Λ(0)e−iϑ have distinct real parts for all phases ϑ ∈ acr(S), and furthermore, the sector S is

sufficiently narrow that there exists a positive real number µ > 0 such that∣∣Re ((λi−λj)e−iϑ

)∣∣ ⩾ µ

for every pair of distinct eigenvalues λi, λj of A(0, 0) and every phase ϑ ∈ arc(S). We will say that

(a germ of) an adjoint orbit o ⊂ g(ℏ) is a generic adjoint orbit if it is the (germ of a) generic

regular semisimple element Λ(ℏ) ∈ g(ℏ).

2. Existence of quantum Levelt filtration.

Theorem 4.11 (quantum Levelt filtration)

Let (E ,∇) be a logarithmic quantum connection germ of rank n at a quantum point p ⊂ X. Assume

the quantum residue Resp∇ is contained in a generic adjoint orbit. Then (E ,∇) is equipped with a

canonical ∇-invariant filtration

E• :=(0 ⊂ E1 ⊂ · · · ⊂ En := E

),

called the quantum Levelt filtration, which has the following two properties:

(1) For any fixed nonzero ℏ ∈ S×, the restriction E (ℏ)• is the Levelt filtration on the germ of a usual

logarithmic connection (E (ℏ)• ,∇(ℏ)) at p(ℏ).

(2) The restriction L(0)• to ℏ = 0 of each associated graded piece L• := E•

/E•−1 is canonically

isomorphic to an eigen-subbundle of the Higgs field ∇(0).

Proof.

We use the local normal form result to build a basis of flat sections of E .

The fundamental solution. Suppose p ⊂ D is a regular singular quantum point of ∇. Choose a

local chart (U;x, ℏ) such that p = x = 0. Then by the regular local normal form result (proposi-

tion 4.10), there exists a local trivialisation φU of E over U in which ∇ has the diagonal form

∇ =loc d− Λ(x)dx

xℏ−K(ℏ)

dx

x,


where Λ0 := Λ(0) has eigenvalues whose real parts are all distinct. In this trivialisation, we see that

E has an obvious fundamental system of ∇-flat sections over U:

Ψ(x, ℏ) := xΛ0/ℏ+K(ℏ)eQ(x)/ℏ , (4.23)

where Q(x) is a diagonal holomorphic matrix5.

The ℏ-family of Levelt filtrations. Now, let λ1(x), . . . , λn(x) be the eigenvalues of Λ(x). Since

∇ is assumed to be nice, the sector S in the ℏ-plane is sufficiently narrow that the eigenvalues

λ1, . . . , λn can be arranged in such an order so that their x-leading order parts λ1(0), . . . , λn(0)

satisfy

Re(λ1(0)/ℏ

)> · · · > Re

(λn(0)/ℏ

)for all ℏ ∈ S×. (4.24)

Let κ1(ℏ), . . . , κn(ℏ) and q1(x), . . . , qn(x) be the corresponding diagonal entries of K and Q. Denote

by (e1, . . . , en) the standard basis of Cn, so that the ei-component of (4.23) is

ψi := xλi/ℏ+κi(ℏ)eqi(x)/ℏei . (4.25)

Then the order in (4.24) implies an order by growth rate at p on the local flat sections ψ1, . . . , ψn

of E over U:

ψ1 ≺ · · · ≺ ψn . (4.26)

Therefore, we obtain a ∇-invariant filtration of E over the open neighbourhood U of p:

E• :=(0 ⊂ E1 ⊂ · · · ⊂ En := E

)where Ei

∼−→φU

⟨ψ1, . . . , ψi⟩OU. (4.27)

The importance of this filtration is the fact that for every fixed nonzero ℏ ∈ S×, the restriction E (ℏ)•

of E• to the fibre X(ℏ) = pr−1(ℏ) is precisely the Levelt filtration of the ordinary regular connection

(E (ℏ),∇(ℏ)) associated to the regular singular point p(ℏ) ∈ D(ℏ).

Thanks to (4.24), the relation (4.26) is valid uniformly for all ℏ ∈ S×, so we have the following

observation. A nice quantum connection with a regular singularity at p specialises at a fixed nonzero

ℏ ∈ S× to an ordinary connection with a regular singularity at p(ℏ), which has the Levelt filtration

associated to p(ℏ), and moreover all these Levelt filtrations fit together uniformly into a smooth

family of filtrations over S×.

5Namely, Q(x) is a primitive of the holomorphic 1-form(Λ− Λ0

)dx /x.


Extension to ℏ = 0. Furthermore, we show that this ℏ-family of Levelt filtrations extends to the

classical curve X(0) = pr−1(0). Indeed, we can use the explicit expression (4.25) for the flat sections

of ∇, and factor out the leading order part in ℏ which is poorly behaved at ℏ = 0. Thus, we equip

E |U with an alternative basis (e1, . . . , en) of generators adapted to the filtration E•, so that:

E• =(0 ⊂ E1 ⊂ · · · ⊂ En := E

)with Ei

∼−→φU

⟨e1, . . . , ei⟩OU. (4.28)

The primary reason that this basis of generators is preferable to (4.25) is the following simple

observation. The trivialisation φU which endows E with this basis of generators is well-defined

over U including the classical fibre U(0) := U∩X(0). This means that the basis (e1, . . . , en) specialises

to a basis (e(0)1 , . . . , e(0)n ) of E (0) over U(0). Moreover, since the growth order e1 ≺ · · · ≺ en holds

uniformly for all ℏ ∈ S×, it induces a corresponding6 order on the generators e(0)1 , . . . , e(0)n . As a

result, the filtration E• of (4.28) extends over the classical fibre X(0).

The spectral eigen-subbundle. Since ψi in (4.25) is a flat section of ∇, it satisfies the equation

∇ψi = 0, which by means of the quantum Leibniz rule can be written as

∇ei = −(λi(0) + ℏκi(ℏ) + xq′i(x)

)dxx

⊗ ei = −(λi(x) + ℏκi(ℏ)

)dxx

⊗ ei .

Restricted to the classical curve X(0) = X|ℏ=0, this identity becomes

−ϕ e(0)i = −λi(x)dx

x⊗ e(0)i . (4.29)

Therefore, the line subbundle⟨e(0)i

⟩O

U(0)⊂ E (0) is an eigen-subbundle of the Higgs field ϕ corre-

sponding to the eigenvalue λi(x) dx /x, defined over U(0).

Consider the graded bundle gr(E•) associated with the filtration E• from (4.28):

gr(E•) :=n⊕

i=1

Li where Li := Ei

/Ei−1.

The basis of generators (e1, . . . , en) for E |U is adapted to the filtration E•, so ei induces a generator

for the i-th graded piece Li, which we continue to denote by ei. In particular, the line bundle L(0)

i

is generated by e(0)i .

6But this order on e(0)1 , . . . , e

(0)n is not an order by growth rate, because the notion of growth rate is not well-defined

on the vector bundle E(0) over the classical fibre X(0), since it is not equipped with a connection.


The spectral eigenbundle L(0) of ϕ is naturally a line bundle on the spectral curve Σ(0) ⊂ T∨X(0)

of ϕ. Let π(0) : Σ(0) → X(0) be the spectral curve of ϕ. Since the open neighbourhood U(0) of the

classical pole p(0) does not contain any classical turning points, let U(0)

i be the sheet of Σ(0) above

U(0) corresponding to the eigenvalue λi(x) dx /x of ϕ. Then the isomorphism π(0) : U(0)

i → U(0) pulls

the line bundle L(0)

i back to a line bundle over U(0)

i , which we continue to denote by L(0)

i . Then

equation 4.29 implies immediately that L(0)

i is the spectral line bundle of ϕ restricted to U(0)

i .

Example 4.12

Consider the rank-2 quantum connection ∇ on the trivial bundle O2X over X := A1

x × S, where S is

the right half-plane, given in the standard basis (e1, e2) for C2 by

∇ := d−

λ1 0

0 λ2

dxxℏ

,

where λ1, λ2 ∈ C are arranged such that λ1 ≺ λ2 (which for a regular singularity — such as the

case in this example — means Re(λ1) > Re(λ2)). Then ∇ has a fundamental basis of flat sections

given by

ψ1 := xλ1/ℏe1 and ψ2 := xλ2/ℏe2 .

Since λ1 ≺ λ2, the resulting regular quantum Levelt filtration on (O2X,∇) associated to the singu-

larity x = 0 is

⟨e1⟩ ⊂ ⟨e1, e2⟩ = O2X .

We can check directly that the associated graded of this filtration restricts to the spectral line bundle

of the associated Higgs field ϕ(0) of ∇:

∇ei = −λiei ⊗dx

xℏgives as ℏ → 0 ϕ(0) ei = λiei ⊗

dx

x.

4.4.1. Nice Quantum Connections


Definition 4.13 (Nice logarithmic quantum connection)

Let (E ,∇) be a logarithmic quantum connection (X,D). We will say that ∇ is nice if it has the

following three properties:

(1) each classical turning point of ∇(0) is simple;

(2) for every component p ⊂ D, the quantum residue Resp∇ lies in a generic adjoint orbit of

g(ℏ);

(3) for every distinct pair p, q ⊂ D and any simply connected subset U ⊂ X containing p, q but no

other component of D, the corresponding quantum Levelt filtrations Ep• , E

q• are transverse.

The subcategory of nice quantum connections will be denoted by QConn∗(X/S,D; gln, oD, a(0)).

When all the fixed data gln, oD, a(0) is clear from the context or unimportant to be explicitly men-

tioned in the discussion, we will often use the following abbreviated notation:

QConnX := QConn∗(X/S,D; gln, oD, a(0)) and QMX := QM∗

dR(X/S,D; gln, oD, a(0)) .

4.5. The WKB Method

4.5.1. The WKB Method

1. The WKB problem. Let X be a smooth complex curve, and let D be an effective divisor on X.

Let (E ,∇) be a connection on (X,D), and suppose E is an extension of OX-modules E ′, E ′′:

0 E ′ E E ′′ 0 . (4.30)

Note that E ′ is not assumed to be ∇-invariant. The central problem of WKB analysis may be stated

as follows: find a splitting W : E ′′ → E of the sequence (4.30) such that W (E ′′) ⊂ E is a ∇-invariant

submodule.

2. The WKB method. The WKB method is a powerful method for solving the WKB problem, and

it can be stated as follows. Suppose we have fixed a pair (W 0,∇0) consisting of a reference splitting

W 0 : E ′′ → E and a reference connection (E ,∇0) on (X,D) such that ∇0 is diagonal with respect to

the decomposition E ∼−→ E ′ ⊕ E ′′ induced by W 0. Using W 0 the sequence (4.30) becomes the split


sequence

0 E ′ E ′ ⊕ E ′′ E ′′ 0 . (4.31)

The essence of the WKB method is to search for a splitting W : E ′′ → E ′⊕E ′′ in the form of a filtered

unipotent gauge transformation of E ′ ⊕ E ′′ with respect to the filtration E ′ ⊂ E ′ ⊕ E ′′. Written in

block-matrix form, any such transformation is

W =

[id S

0 id

]:

E ′ E ′

E ′′ E ′′⊕ ⊕ , (4.32)

for some map S : E ′′ → E ′. Then we write ∇ in terms of the reference connection ∇0:

∇ = ∇0 − ϕ

for some twisted endomorphism ϕ ∈ End(E)⊗ Ω1X(D), written in block-matrix form as

ϕ =

[ϕ11 ϕ12ϕ21 ϕ22

]:

E ′ E ′

E ′′ E ′′⊕ ⊕ ⊗ Ω1

X(D) . (4.33)

Next, we apply the gauge transformation W to ∇ = ∇0−ϕ. First, to gauge transform ∇0, we write

it in block-diagonal form: ∇0 = ∇′ ⊕∇′′. If e′ ∈ E ′, e′′ ∈ E ′′, then we compute:

W−1 ∇0 W

[e′

e′′

]=

[id −S0 id

][∇′ 0

0 ∇′′

][id S

0 id

][e′

e′′

]

=

[∇′e′ +∇′(S(e′′))− S(∇′′e′′)

∇′′e′′

]

=

[∇′e′ + [∇0, S](e′′)

∇′′e′′

]

=

(∇0 +

[0 [∇0, S]

0 0

])[e′

e′′

],

where [∇0, S] = ∇0S − S∇0 : E ′′ → E ′ ⊗ Ω1X(D) is the commutator. Thus, as a result, applying the

gauge transformation W to ∇ = ∇0 − ϕ, we find

W−1 ∇W = ∇0 +

[0 [∇0, S]

0 0

]−

[ϕ11−S ϕ21 ϕ11 S − S ϕ21 S + ϕ12−S ϕ22

ϕ21 ϕ21 S + ϕ22

].


If the map S : E ′′ → E ′ can be chosen such that

[∇0, S]− ϕ11 S + S ϕ21 S − ϕ12+S ϕ22 = 0 , (4.34)

then the gauge transformed connection W−1 ∇W becomes lower-triangular:

W−1 ∇W = ∇0 −

[ϕ11−S ϕ21 0

ϕ21 ϕ21 S + ϕ22

].

This means that if ψ ∈ E ′′ is a flat section for the connection

∇0 − ϕ21 S − ϕ22 : E ′′ −→ E ′′ ⊗ Ω1X(D) , (4.35)

then Wψ = ψ + Sψ is a flat section for ∇.

3. The WKB splitting. We axiomatise the discussion of the previous paragraph. Let (E ,∇) be a

connection on (X,D), and suppose E is a split extension of OX-modules E ′, E ′′:

0 E ′ E E ′′ 0 . (4.36)

Suppose we are given a reference connection ∇0 on E over (X,D) which is diagonal with respect to

the decomposition E ∼−→ E ′ ⊕ E ′′ induced by the given splitting of (4.36), and write

∇0 −∇ =: ϕ =

[ϕ11 ϕ12ϕ21 ϕ22

]:

E ′ E ′

E ′′ E ′′⊕ ⊕ ⊗ Ω1

X(D) .

Definition 4.14 (generalised Riccati equation)

The equation

[∇0, S]− ϕ11 S + S ϕ21 S − ϕ12+S ϕ22 = 0

for S : E ′′ → E ′ is called the generalised Riccati equation.


Definition 4.15 (WKB splitting)

A WKB splitting is a filtered unipotent splitting of (4.36) of the form

W =

[id S

0 id

]:

E ′ E ′

E ′′ E ′′⊕ ⊕ ,

where S satisfies the Riccati equation.

Taking these definitions into account, the discussion in paragraph 2 can be summarised as follows.

Theorem 4.16 (the WKB ansatz)

If W : E ′′ → E is a WKB splitting, then W (E ′′) ⊂ E is a ∇-invariant submodule.

4.5.2. The Schrodinger Equation

The WKB method especially shines when applied to quantum connections, because powerful meth-

ods of perturbation theory may then be brought into the picture. We demonstrate our point of view

on the WKB method through the example of the Schrodinger equation.

1. Quantum spin structure. Recall that ωX/S denotes the relative canonical bundle on X. Local

sections of ωX/S are differentials of the form f(x, ℏ) dx. Choose a square root ω1/2X/S; by definition,

ω1/2X/S is a line bundle over X together with an isomorphism

κ : ω1/2X/S ⊗ ω

1/2X/S

∼−→ ωX/S .

In general, a square root of the canonical bundle on a curve is often referred to as the spin structure

on X [Ati71, section 3]. By analogy, we shall refer to ω1/2X/S as the quantum spin structure on X.

Given a local coordinate (U;x, ℏ) on X, the OX-module ω1/2X/S has a canonical generator over U

denoted by dx1/2 (called adapted trivialisation), which has the property that κ : dx

1/2 ⊗ dx1/2 7→ dx.

If y = g(x, ℏ) is another coordinate, then

dy1/2 = (∂xg)

1/2 dx1/2 ;

notice that the square root (∂xg)1/2 is well-defined since ∂xg ∈ O×

X


2. Quantum Anti-Spin Structure and Its Quantum Jets. Now, consider the line bundle dual to

the quantum spin structure ω1/2X/S:

L := ω−1/2X/S .

Given a local coordinate (U;x, ℏ) on X, the line bundle L similarly has a canonical generator over

U, denoted by dx−1/2, defined by the property that dx−1/2 ⊗ dx

1/2 = 1.

Consider the quantum 1-jet bundle of L:

E := J 1ℏ L . (4.37)

The corresponding quantum jet sequence is

0 Ω1,ℏX/S(L) E L 0 ,

The vector bundle E is an extension of L by L∨ ⊗ OX(C). Such extensions are classified by

Ext1(L,L∨(C)) ∼= H1

X

(Ω1,ℏX/S

),

where we used the isomorphism L∨ ⊗ L∨ ∼−→ ωX/S = Ω1X/S.

The OX-module E is locally free of rank 2. If (U;x, ℏ) is a local coordinate on X, then E |U is

generated by dx−1/2 and ℏ−1 dx⊗ dx−1/2 = dx1/2/ℏ.

To sum up, a choice of a local coordinate (U;x, ℏ) on X equips the bundles we consider with adapted

local generators over U as follows:

ωX/S∼−→ ⟨dx⟩, L def= ω

−1/2X/S

∼−→ ⟨dx−1/2⟩, Ω1,ℏU/S

∼−→ ⟨dx /ℏ⟩ ,

E ∼−→⟨dx−

1/2 , ℏ−1dx⊗ dx−1/2⟩

.

If ψ = f dx−1/2 ∈ L is a local section, where f = f(x, ℏ), then

j1ℏψ = f dx−1/2 + ℏ∂xf dx

1/2/ℏ .

Denote ℓ := dx−1/2 and e := ℏ−1 dx⊗ dx−

1/2 = ℏ−1 dx⊗ ℓ.


3. Connections on Quantum Jets. An adapted trivialisation of E |U equips it with a natural quan-

tum connection, the relative de Rham differential:

d : E |U −→ Ω1,ℏU/S ⊗ E |U ,

which in the basis (ℓ, e) acts diagonally:

d(fℓ)= ℏ∂xf dx/ℏ⊗ ℓ, d

(fe)= ℏ∂xf dx/ℏ⊗ e ,

for any f ∈ OX. This will be the reference connection ∇0 needed to apply the WKB method.

Any other quantum connection ∇ on E differs from d by an Ω1U/ℏ-valued endomorphism of E:

ϕ := d−∇ ∈ Ω1U/ℏ(End E) .

Written in the basis (ℓ, e), ϕ has the form

ϕ =

ϕ11 ϕ12

ϕ21 ϕ22

,

where ϕ11, ϕ22 ∈ Ω1,ℏU/S, ϕ12 ∈ OU, and ϕ21 ∈ Sym2Ω1,ℏ

U/S.

4. The Schrodinger Connection. A quantum connection ∇ on the quantum jet bundle E = J 1ℏ L

of a quantum anti-spin bundle L = ω−1/2X/S has two additional important features. First, it is a

connection on an extension of line bundles:

0 Ω1,ℏX/S(L) E L 0

(0 Ω1,ℏ

X/S(L) E L 0)⊗ Ω1,ℏ

X/S .

∇

As such, the induced determinant connection det∇ on the corresponding determinant bundle of

E is easy to compute: since det E is Ω1,ℏX/S(L

2) ∼−→κ OX, the determinant connection is in fact a

quantum connection on the structure sheaf OX.

det∇ : OX −→ Ω1,ℏX/S .


Second, once again thanks to the quantum jet sequence, ∇ is a connection on a filtered bundle

Ω1,ℏX/S(L) −→ E = E• .

The quotient E/E1 is naturally isomorphic to L, so the associated graded gr E• is naturally isomor-

phic to Ω1,ℏX/S(L)⊕ L. The connection ∇ induces an OX-linear map

gr∇ : gr E• −→ Ω1,ℏX/S ⊗ gr E• .

Its (12)-piece gr12∇ : gr1 E• −→ Ω1X/ℏ ⊗ gr2 E• is a function, since gr1 E• = E1 and gr2 E• = L. This

function measures the failure of ∇ to preserve the filtration E•.

Definition 4.17 (Schrodinger connection)

A Schrodinger connection is a quantum connection ∇ on E = J 1ℏ (ω

−1/2X/S ) such that

(1) det∇ = d : OX −→ A∨X/S,

(2) gr12∇ = 1 ∈ OX.

5. The WKB method. Let D ⊂ X be a quantum divisor, let AX/S := T ℏX/S(− logD). Let (E ,∇)

be a nice logarithmic Schrodinger connection on (X,D). Then E = J 1ℏ L fits into the quantum jet

sequence

0 A∨X/S ⊗ L E L 0 , (4.38)

Choose a local coordinate (U;x, ℏ) on X, which canonically selects the following trivialisations over

U:

ωX/S∼−→ ⟨dx⟩, ω

−1/2X/S

∼−→ ⟨dx−1/2⟩, Ω1,ℏU/S

∼−→ ⟨dx /ℏ⟩ ,

E ∼−→⟨dx−

1/2 , ℏ−1dx⊗ dx−1/2⟩

.

The choice of (U;x, ℏ) determines a reference splitting of (4.38) and a reference connection ∇0 := d

as required in order to apply the WKB method.

Let G be an unipotent filtered gauge transformation on E with respect to the filtration A∨X/S(L) →

E . Denote ℓ := dx−1/2, and e := ℏ−1dx ⊗ dx−1/2 = ℏ−1dx⊗ ℓ. Then in the basis (ℓ, e) for E |U, the


gauge tranformation G has the following form

G =loc

[1 0

S 1

],

where 1 ∈ OU and S ∈ L∨ ⊗ A∨X/S(L) ∼−→

κΩ1,ℏU/S. Writing S = s(x, ℏ) dx /ℏ, we have S(ℓ) = se. In

the basis (ℓ, e), the quantum connection ∇ has the form

∇ =loc d−

[0 1

q 0

]dx

ℏ.

Applying the gauge transformation G to ∇, we find:

G−1∇Gℓ = sdxℏ ⊗ ℓ+(ℏ∂xs+ s2 − q

)dxℏ ⊗ e .

Thus, if ℏ∂xs + s2 = q, then the gauge transformation G defines a ∇-invariant subbundle G(L).

Therefore, the generalised Riccati equation (4.34) in this case reduces to the familiar Riccati equa-

tion:

ℏ∂xs+ s2 = q . (4.39)

This Riccati equation is most easily solved in formal power series in ℏ; i.e., by means of perturbation

theory techniques. This will be demonstrated in section 5.3.

4.5.3. Opers and Transverse Extensions

The WKB method is especially convenient in application to opers. Our definitions in this section

are straightforward generalisations of definitions found in [Sim10, §2]. Let X → S be a quantum

family, and (E ,∇) a quantum connection on (X,D).

Definition 4.18 (Griffiths transverse filtration)

A Griffiths transverse filtration on E is a filtration

E• :=(0 = E0 ⊂ E1 ⊂ · · · ⊂ Em := E

),

satisfying the Griffiths transversality condition with respect to ∇:

∇(Ek) ⊂ Ek+1 ⊗ Ω1X/S(D) .


Given a quantum connection (E ,∇) on (X,D) with a Griffiths transverse filtration E•, consider the

associated vector graded bundle

Definition 4.19 (oper)

An oper on (X,D) is a meromorphic connection (E ,∇) on (X,D) equipped with a full Griffiths

transverse filtration E• such that each induced OX-linear map

grk ∇ : Lk −→ Lk+1 ⊗ Ω1X/S(D)

is an isomorphism.

1. The jet filtration. Let P ∈ DL(X,D) be a logarithmic differential operator on a line bundle L.

Then by means of the jet functor, P induces a connection ∇ on the jet bundle J kL, where k is

the order of P . The connection ∇ is necessarily an oper: its filtration is known as the jet filtration,

which we describe in slightly greater generality.

Let V be a vector bundle on X. The n-th jet bundle E := J nV is carries a natural filtration as

follows. Let Ei be the kernel of the canonical i-th jet projection map:

Ei := ker(

J nV −→ J n−iV)

.

Then evidently Ei ⊂ Ei+1. The resulting filtration

(J nV

)• := E• :=

(0 =: E0 ⊂ E1 ⊂ · · · ⊂ En+1 := E

)will be referred to as the jet filtration. The associated graded of the jet filtration is easy to compute

using the jet sequence. Indeed, E1 is just the kernel of the n-th jet sequence, and in fact more

generally:

Ei+1

/Ei = Symn−i A∨ ⊗ V .

Therefore,

gr((

J nV)•

)=

n⊕i=0

Symn−i A∨ ⊗ V .

The jet filtration generalises in a straightforward manner to the quantum jet filtration on quantum

jet bundles.

chapter 5

Abelianisation

of Quantum sl2-Connections

Contents5.1 The Abelianisation Functor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122

Construction of (Eab,∇ab). . . . . . . . . . . . . . . . . . . . . . . . . 123

Properties. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124

Extension of spectral line bundle. . . . . . . . . . . . . . . . . . . . . 125

5.2 Quantum Abelianisation Equivalence . . . . . . . . . . . . . . . . . . . . . . . . 125

5.3 Abelianisation and the Exact WKB Method . . . . . . . . . . . . . . . . . . . . . 126

121


5.1. The Abelianisation Functor

Let X be a quantum family, and D a reduced effective quantum divisor on X. Fix generic quan-

tum sl2-residue data oD, as well as a nice saddle-free quadratic differential q(0) ∈ H1C

(ω2C(2D

(0))).

Consider the category

QConnX := QConn∗(X/S,D; sl2, oD, q(0))of nice quantum connections with fixed quantum residue data oD and classical spectral data q(0).

In this section, we generalise the abelianisation functor of section 3.3 to quantum sl2-connections.

The domain of this functor is the category QConnX of nice quantum connections on X, and its

codomain likewise depends on an additional choice of a quadratic differential q ∈ ω2X/S(2D) com-

patible with the fixed residue and classical spectral data in the following sense.

Definition 5.1 (compatibility between q and quantum oD)

A quadratic differential q ∈ H0(X, ω2

X/S(2D))

is said to be compatible with the quantum residue

data oD and the classical quadratic differential q(0) ∈ H0C

(ω2C(2D

(0)))

if

(1) q extends q(0); i.e., q∣∣ℏ=0

= q(0);

(2) Res2D q = det(ResD∇) for any ∇ ∈ QConnX.

Such a quadratic differential q ∈ ω2X/S(2D) determines a surface Σ := Zero(η2taut − q) sitting inside

the total space of the twisted quantum cotangent sheaf A∨X/S:

Σ tot(A∨

X/S

)X .

ι

π π

Notice that restriction of Σ to ℏ = 0 is the classical spectral curve Σ(0). In this sense, the surface Σ

extends the classical spectral curve; we shall refer to it as the quantum spectral curve.

Denote by η the pullback to Σ of ηtaut. Notice again that η∣∣ℏ=0

= η(0). On each fibre Σ(ℏ), the 1-

form η(ℏ) represents the canonical square root of the quadratic differential q(ℏ). As such, the Levelt

open cover UΣ from section 3.3 has a straightforward generalisation to an open cover QUΣ of

Σ× := Σ \ R, which we will call the quantum Levelt open cover of Σ×. Its open sets are cylinders

Upi , called quantum Levelt cells, each slice Up,(h)

i being a Levelt cell on Σ(ℏ).


Let oabD be the abelianisation of the quantum residue data, and consider the category

QConnΣ := QConnodd(Σ×/S, π∗D; ab, oabD , η

(0))

of odd logarithmic quantum abelian connections on the punctured quantum spectral curve Σ× =

Σ \ R with poles along π∗D, fixed quantum residue data oabD , and fixed classical spectral data η(0).

Definition 5.2 (abelianisation functor)

The functor

πab : QConnX −→ QConnΣ

(E ,∇) 7−→ (Eab,∇ab) .

constructed in this section will be called the abelianisation functor. The abelian quantum con-

nection (Eab,∇ab) will be called the abelianisation quantum connection associated with (E ,∇)

and q.

1. Construction of (Eab,∇ab). Let X be a quantum family, and let D be a reduced effective quan-

tum divisor on X. Let a nice quantum connection (E ,∇) ∈ QConn∗(X/S,D; sl2, oD;ϕ(0)) and a nice

saddle-free quadratic differential q ∈ H0X

(ω2X/S(2D)

)compatible with the quantum residue data oD

and classical spectral data q(0) be given. We use the quantum Levelt open covering of Σ× to give a

Cech construction of an abelian quantum connection (Eab,∇ab) in the category QConnΣ.

Let p ⊂ D, and let Up1,U

p2 be the Levelt cells associated with the two preimages p1, p2 ⊂ π∗D of p.

Let Up be the corresponding quantum Levelt cell on the base X. By theorem 4.11, the quantum

connection (E ,∇) is naturally equipped over Up with the quantum Levelt filtration corresponding

to p:

Ep• :=

(0 ⊂ Lp

1 ⊂ E)

.

Consider the associated graded bundle, defined over Up:

gr(Ep• ) = Lp

1 ⊕ Lp2 where Lp

2 := E/

Lp1 .

Since the quantum Levelt line Lp1 is ∇-invariant, ∇ induces abelian quantum connections ∇p

1,∇p2


on each Lp1,L

p2:

∇p1 : Lp

1 −→ Lp1 ⊗ A∨

U/S and ∇p2 : Lp

2 −→ Lp2 ⊗ A∨

U/S .

Thus, for every quantum Levelt cell Upi ∈ QUΣ, we define:

(Eab,∇ab)∣∣Upi

:= (Lpi ,∇

pi ) .

So let U be a quantum Stokes region, so it is (a connected component of) the double intersection

Upi∩U

qj of two quantum Levelt cells. Then π∗E over U is naturally equipped with two quantum Levelt

filtrations Ep• , E

q• , and these filtrations are transverse. Then just as in the case of usual connections,

transversality induces the required canonical gluing maps.

The quantum Levelt open covering QUΣ contains no triple intersections so the construction of

(Eab,∇ab) over Σ× is complete.

2. Properties. Just as in the case of usual connections (section 3.3.2), the vector bundle π∗Eab is

canonically represented over each quantum Levelt cell Up ∈ QUX on the base X by the associated

graded vector bundle gr(Ep• ). This observation will be useful to us later.

Proposition 5.3

The abelianisation quantum connection (Eab,∇ab) is odd and has monodromy −1 around the ramifi-

cation locus. Furthermore, the correspondence (E ,∇) 7−→ (Eab,∇ab) extends to a functor

πab : ConnX −→ ConnΣ .

Proof.

The proof of these statements follows the exact same line of logic as the proofs of corresponding

statements in section 3.3.2.

All these properties of abelianisation quantum connections are direct generalisations of the corre-

sponding properties for usual connections. Let us now discuss a new phenomenon that arises by

virtue of the fact that quantum connections are particular ℏ-families of usual connections.


3. Extension of spectral line bundle. Locally on Σ×, the abelianisation line bundle is canonically

represented by a piece Lpi of the associated graded of a quantum Levelt filtration Ep

• of E . By part

(2) of theorem 4.11, the restriction Lp,(0)i = Lp

i

∣∣ℏ=0

of Lpi to the punctured classical spectral curve

(Σ(0))× = Σ(0) \ R(0) is canonically isomorphic to the spectral line bundle L(0) of the classical Higgs

field ∇(0). This proves the following theorem.

Theorem 5.4

The restriction of the abelianisation quantum connection (Eab,∇ab) to the punctured classical spectral

curve (Σ(0))× of the Higgs bundle (E (0),∇(0)) is canonically isomorphic to (L(0), η(0)) where L(0) is the

spectral line bundle of ∇(0).

5.2. Quantum Abelianisation Equivalence

Theorem 5.5

The abelianisation functor

πab : QConnX QConnΣ∼

is an equivalence of categories.

Proof.

Choose a nice quantum connection (E ,∇) ∈ QConnX. Let (Eab,∇ab) ∈ QConnΣ be its abelianisa-

tion, and consider (π∗Eab, π∗∇ab) ∈ QConn×X . Consider the quantum Stokes open cover UX of X×.

The construction in the proof of proposition 3.28 readily generalises to the situation of quantum

connections. Thus, we obtain a canonical Cech 1-cocycle v(∇ab) ∈ Z1(SX,Aut(π∗Eab)

), which also

admits a description in terms of the path-lifting rule as in proposition 3.32. But the parallel trans-

port of ∇ab along the detour path exhibits an interesting behaviour as follows. Since (∇ab)(0) = η(0),

we see that ∇ab = d+ η(0)/ℏ+O(ℏ). At the same time, the integral of η(0)

/ℏ along a detour path δ

has a positive real part. The parallel transport of ∇ab is of the form exp(−∫δ η

(0)/ℏ+O(ℏ)

)which

vanishes in the limit ℏ → 0 in S. As a result, ∆(ℏ)|ℏ=0 = 0, and so the canonical cocycle v(∇ab) is

trivial on the classical fibre C = X(0).

Thanks to the interpretation in terms of the path-lifting rule, just like in section 3.4.3, we deduce

that this procedure defines a Cech 1-cocycle v ∈ Z1(SX,Aut(π∗)

), which deforms the pushforward

functor π∗ to give a new functor πv∗ : QConnΣ → QConn×X which factors through QConnX. The

resulting functor πvab : QConnΣ → QConnX is an inverse equivalence to the abelianisation functor


πab essentially by construction.

5.3. Abelianisation and the Exact WKB Method

Consider the logarithmic Schrodinger connection (E ,∇) on the quantum jet bundle E := J 1ℏ L of

a quantum anti-spin bundle L := ω−1/2X/S with poles along D. In section 4.5.2, we applied the WKB

method to ∇. For this, we chose a local coordinate (U;x, ℏ) on X, which determined the following

canonical trivialisations over U:

ωX/S∼−→ ⟨dx⟩, ω

−1/2X/S

∼−→ ⟨dx−1/2⟩, A∨X/S

∼−→ ⟨dx /ℏ⟩ ,

E ∼−→⟨dx−

1/2 , ℏ−1dx⊗ dx−1/2⟩

.

In this trivialisation,

∇ =loc d−

[0 1

q 0

]dx

ℏ

and a WKB splitting is given by

G =loc

[1 0

S 1

],

where S = s(x, ℏ) dx /ℏ and s(x, ℏ) satisfies the Riccati equation:

ℏ∂xs+ s2 = q . (5.1)

Theorem 5.6 (Koike-Schafke [KS])

For each polar divisor pi ⊂ π∗D ⊂ Σ, the Riccati equation (5.1) has a unique solution s = s(x, ℏ) in

U with a simple pole at pi.

We provide only a brief sketch of the proof1.

Idea of the Proof.

To find a solution to the Riccati equation (5.1), we begin by constructing a formal solution of the

form

s(x, ℏ) :=∞∑k=0

ak(x)ℏk = a0 + a1ℏ+ a2ℏ2R

1The author is grateful to Kohei Iwaki for sharing his notes which so clearly explain Koike-Schafke’s proof of thistheorem.


for some R = R(x, ℏ). The first three orders in ℏ of (5.1) are:

ℏ0 | a20 = q0 ;

ℏ1 | a′0 + 2a0a1 = q1 ;

ℏ2 | a′1 + a21 + 2a0a2 = q2 .

We use these identities to rewrite (5.1) as an ODE for Rℏ, to which we apply the formal Borel

transform B in the variable ℏ to obtain a first order PDE for τ = τ(x, t), where τdt := B(Rℏ).

For any compactly contained open subset V ⋐ U(0), we can then solve this PDE iteratively on the

domain V(0) ×W, where W is an open neighbourhood of the positive real axis R+ ⊂ Ct; moreover,

the solution τ is holomorphic in V×W and satisfies the following exponential estimate: there exist

constants C1, C2 > 0 such that for any (x, t) ∈ V ×W,

∣∣τ(x, t)∣∣ ⩽ C1eC2|t| .

Subsequently, the ODE for Rℏ has a holomorphic solution Rℏ = R(x, ℏ)ℏ on V, given as the Laplace

transform of τdt:

Rℏ :=

+∞∫0

e−t/ℏτ dt .

Then

s(x, ℏ) := a0 + a1ℏ+ a2ℏ2R

is a solution to the Riccati equation (5.1).

Corollary 5.7

For any pi ∈ π∗D, let Upi ⊂ Σ0 be the corresponding Levelt cell. Then, over Up

i , there exists a WKB

splitting W such the ∇-invariant subbundle W(L)

is naturally isomorphic to the piece Lpi of the

associated graded gr(Ep• ) of the quantum Levelt filtration Ep

• .

In other words, the WKB basis of E over the Levelt cell Up ⊂ X is an adapted basis to the quantum

Levelt filtration Ep• .

Appendix

ContentsA Spectral Splitting Convergence . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129

A.1 Spectral Splitting | Convergence of the formal solution . . . . . . . . . . . 129

A.2 Quantum Spectral Splitting | Power Series is Gevrey . . . . . . . . . . . . . 130

128


A. Spectral Splitting Convergence

A.1. Spectral Splitting | Convergence of the formal solution

Jump back to paragraph 2

We show that the power series G is in fact convergent. Since A(0)(x) is holomorphic at x = 0, the

coefficients in Aijk in the power series for Aij(x) (equation 4.10) have a bound: there exist some

positive real constants C,M such that

∥∥∥Aijk

∥∥∥ ⩽ CMk for all k ⩾ 0 . (2)

The lefthand side of equation 4.12 is linear in the entries of the matrix Gk, so it can be written in

the form A0Gk, where A0 is a nonsingular matrix whose entries are linear combinations of entries

of A110 , A

220 ; and Gk is a column vector whose entries are the entries of Gk (in some unspecified

order). Thus, if Ak is a column vector such whose entries are the entries of the righthand side of

equation 4.12 (in the same order as for Gk), then Gk = A−10 Ak. Therefore there exists a positive

constant C0 such that

∥Gk∥ ⩽ C0

∥∥∥∥(righthand side of (4.12))∥∥∥∥ . (3)

Suppose ∥Gk∥ ⩽ gkMk for some positive real numbers gk. Then (3) gives:

gkMk ⩽ C0

(CMk +

k−1∑s=1

(CMkgs + CMkgk−s

)+

k−1∑s=1

k−s−1∑m=1

CMkgmgs

)

= CC0Mk

(1 +

k−1∑s=1

(gs + gk−s) +k−1∑s=1

k−s−1∑m=1

gmgs

).

Thus, let us examine the auxiliary sequence of positive real numbers given by

gk := CC0

(1 +

k−1∑s=1

(gs + gk−s) +

k−1∑s=1

k−s−1∑m=1

gmgs

),

and g1 := K for some positive real constant K to be determined shortly. Proceeding by induction,


assume that gs ⩽ Ks for all s < k. Then, using the formula for the geometric series, we estimate:

gk ⩽ CC0

(1 +

k−1∑s=1

(Ks +Kk−s) +

k−1∑s=1

k−s−1∑m=1

Km+s

)

= CC0

(1 +Kk

k−1∑s=1

K−s +k−1∑s=1

Ks

(1 +

k−s−1∑m=1

Km

))

⩽ CC0

(1 +Kk

∞∑s=1

K−s +∞∑s=1

Ks

(1 +

∞∑m=1

Km

))

= CC0

(1 +Kk K−1

1−K−1+

∞∑s=1

Ks

(1 +

K−1

1−K−1

))

= CC0

(1 +Kk K−1

1−K−1+

K−1

(1−K−1)2

)⩽ CC0K

k

(K−k +

K−1

1−K−1+

K−(k+1)

(1−K−1)2

).

By choosing K large enough, the quantity in the bracket can be made smaller than (CC0)−1 for all

k ⩾ 1, proving that gk ⩽ Kk. Since gk ⩽ gk and ∥Gk∥ ⩽ gkMk, we conclude that

∥Gk∥ ⩽ (KM)k for all k ⩾ 0 ,

which proves that G is holomorphic at x = 0.

A.2. Quantum Spectral Splitting | Power Series is Gevrey

Jump back to paragraph 2

By assumption, A(x, ℏ) is uniformly Gevrey in the sector S, which means that there exist positive

real constants C,M such that the expansion coefficients in (4.19) have the following bounds2:

∥∥∥A(k)

ij

∥∥∥ ⩽ CMkk! .

This implies that G(x, ℏ) is Gevrey power series in ℏ, as we now show.

By the exact same argument as in the paragraph immediately following (2), we deduce that there

2Take C,M sufficiently large that they do not depend on i, j


exists a positive constant C0 (independent of k) such that

∥Gk∥ ⩽ C0

∥∥∥∥(righthand side of (4.21))∥∥∥∥ , (4)

uniformly for all x ∈ U. Suppose ∥Gk∥ ⩽ gkMkk! for some positive real numbers gk. Then the

estimate (4) gives:

gkMkk! ⩽ C0

(CMkk! +

k−1∑s=1

(CMkgs(k − s)!s! + CMkgk−s(k − s)!s!

)+

k−1∑s=1

k−s−1∑m=1

CMkgmgsm!s!(k −m− s)!

)

= C0CMkk!

(1 +

k−1∑s=1

(k − s)!s!

k!(gs + gk−s) +

k−1∑s=1

k−s−1∑m=1

(k −m− s)!m!s!

k!gmgs

)

⩽ C0CMkk!

(1 +

k−1∑s=1

(gs + gk−s) +

k−1∑s=1

k−s−1∑m=1

gmgs

),

where we used the fact that (k−s)!s! ⩽ k! and (k−m−s)!m!s! ⩽ k!. We have already encountered

the sequence of real numbers defined recursively by the expression in the brackets; we found that

there is a positive real number K such that gk ⩽ Kk. Thus, we conclude that the ℏ-formal power

series G is uniformly Gevrey in U:

∥Gk∥ ⩽ (KM)kk! .

Bibliography

[AB83] M. F. Atiyah and R. Bott, “The Yang-Mills equations over Riemann surfaces”, Philos.Trans. Roy. Soc. London Ser. A 308 no. 1505, (1983) 523–615.

[AKT91] T. Aoki, T. Kawai, and Y. Takei, “The Bender-Wu analysis and the Voros theory”, inSpecial functions (Okayama, 1990), ICM-90 Satell. Conf. Proc., pp. 1–29. Springer,Tokyo, 1991.

[Ati71] M. F. Atiyah, “Riemann surfaces and spin structures”, Ann. Sci. Ecole Norm. Sup. (4) 4(1971) 47–62.

[BNR89] A. Beauville, M. S. Narasimhan, and S. Ramanan, “Spectral curves and the generalisedtheta divisor”, J. Reine Angew. Math. 398 (1989) 169–179.

[Boa99] P. Boalch, Symplectic Geometry and Isomonodromic Deformations. PhD thesis, Universityof Oxford, 1999. https://www.math.u-psud.fr/~boalch/thesis/.

[Boa01] P. Boalch, “Symplectic manifolds and isomonodromic deformations”, Adv. Math. 163no. 2, (2001) 137–205.

[BS15] T. Bridgeland and I. Smith, “Quadratic differentials as stability conditions”, Publ. Math.Inst. Hautes Etudes Sci. 121 (2015) 155–278.

[Cos09] O. Costin, Asymptotics and Borel summability, vol. 141 of Chapman & Hall/CRC Mono-graphs and Surveys in Pure and Applied Mathematics. CRC Press, Boca Raton, FL, 2009.

[DDP93] E. Delabaere, H. Dillinger, and F. Pham, “Resurgence de Voros et periodes des courbeshyperelliptiques”, Ann. Inst. Fourier (Grenoble) 43 no. 1, (1993) 163–199.

[DH75] A. Douady and J. Hubbard, “On the density of Strebel differentials”, Invent. Math. 30no. 2, (1975) 175–179.

[DM14] O. Dumitrescu and M. Mulase, “Quantization of spectral curves for meromorphic higgsbundles through topological recursion”, arXiv:1411.1023 [math.AG].

[FG06] V. Fock and A. Goncharov, “Moduli spaces of local systems and higher Teichmullertheory”, Publ. Math. Inst. Hautes Etudes Sci. no. 103, (2006) 1–211, arXiv:0311149[math.AG].

[GLP15] M. Gualtieri, S. Li, and B. Pym, “The stokes groupoids”, Journal fur die reine und ange-wandte Mathematik (Crelles Journal) (2015) . arXiv:1305.7288.

[GMN13a] D. Gaiotto, G. W. Moore, and A. Neitzke, “Spectral networks”, Ann. Henri Poincare 14no. 7, (2013) 1643–1731, arXiv:1204.4824 [hep-th].

[GMN13b] D. Gaiotto, G. W. Moore, and A. Neitzke, “Wall-crossing, Hitchin systems, and the WKBapproximation”, Adv. Math. 234 (2013) 239–403, arXiv:0907.3987 [hep-th].

[Gro61] A. Grothendieck, “Elements de geometrie algebrique. II. Etude globale elementaire dequelques classes de morphismes”, Inst. Hautes Etudes Sci. Publ. Math. no. 8, (1961)222.

132

http://dx.doi.org/10.1098/rsta.1983.0017

http://dx.doi.org/10.1098/rsta.1983.0017

http://dx.doi.org/10.24033/asens.1205

http://dx.doi.org/10.24033/asens.1205

http://dx.doi.org/10.1515/crll.1989.398.169

https://www.math.u-psud.fr/~boalch/thesis/

http://dx.doi.org/10.1006/aima.2001.1998

http://dx.doi.org/10.1006/aima.2001.1998

http://dx.doi.org/10.1007/s10240-014-0066-5

http://dx.doi.org/10.1007/s10240-014-0066-5

http://dx.doi.org/10.1007/BF01425507

http://dx.doi.org/10.1007/BF01425507

http://arxiv.org/abs/1411.1023

http://dx.doi.org/10.1007/s10240-006-0039-4

http://arxiv.org/abs/0311149

http://arxiv.org/abs/0311149

arXiv:1305.7288

http://dx.doi.org/10.1007/s00023-013-0239-7

http://dx.doi.org/10.1007/s00023-013-0239-7


http://dx.doi.org/10.1016/j.aim.2012.09.027


[GW16] S. Gupta and M. Wolf, “Meromorphic quadratic differentials with complex residues andspiralling foliations”, arXiv:1607.06931.

[Hea62] J. Heading, An introduction to phase-integral methods. Methuen & Co., Ltd., London;John Wiley & Sons, Inc., New York, 1962.

[Hit87] N. J. Hitchin, “The self-duality equations on a riemann surface”, Proceedings of theLondon Mathematical Society s3-55 no. 1, (1987) 59–126.

[HM79] J. Hubbard and H. Masur, “Quadratic differentials and foliations”, Acta Math. 142 no. 3-4, (1979) 221–274.

[HN16] L. Hollands and A. Neitzke, “Spectral networks and Fenchel-Nielsen coordinates”, Lett.Math. Phys. 106 no. 6, (2016) 811–877.

[HP12] T. Hausel and C. Pauly, “Prym varieties of spectral covers”, Geom. Topol. 16 no. 3,(2012) 1609–1638.

[HS99] P. Hsieh and Y. Sibuya, Basic Theory of Ordinary Differential Equations. Univer-sitext (1979). Springer New York, 1999. https://books.google.ca/books?id=

pMyEmNBkyBsC.

[HSW99] N. J. Hitchin, G. B. Segal, and R. S. Ward, Integrable systems, vol. 4 of Oxford Grad-uate Texts in Mathematics. The Clarendon Press, Oxford University Press, New York,1999. Twistors, loop groups, and Riemann surfaces, Lectures from the InstructionalConference held at the University of Oxford, Oxford, September 1997.

[IN14] K. Iwaki and T. Nakanishi, “Exact WKB analysis and cluster algebras”, J. Phys. A 47no. 47, (2014) 474009, 98.

[Kor04] D. Korotkin, “Solution of matrix Riemann-Hilbert problems with quasi-permutationmonodromy matrices”, Math. Ann. 329 no. 2, (2004) 335–364.

[KS] T. Koike and R. Schafke, “On the Borel summability of WKB solutions of Schrodingerequations with polynomial potentials and its application”, in preparation .

[KT05] T. Kawai and Y. Takei, Algebraic analysis of singular perturbation theory, vol. 227 ofTranslations of Mathematical Monographs. American Mathematical Society, Providence,RI, 2005. Translated from the 1998 Japanese original by Goro Kato, Iwanami Series inModern Mathematics.

[Lev61] A. H. M. Levelt, Hypergeometric functions. Doctoral thesis, University of Amsterdam.Drukkerij Holland N. V., Amsterdam, 1961.

[MP98] M. Mulase and M. Penkava, “Ribbon graphs, quadratic differentials on Riemann sur-faces, and algebraic curves defined over Q”, Asian J. Math. 2 no. 4, (1998) 875–919.

[Pym13] B. Pym, Poisson structures and Lie algebroids in complex geometry. PhD thesis, Universityof Toronto, 2013.

[Sab] C. Sabbah, “Introduction to algebraic theory of linear systems of differential equations.”http://www.cmls.polytechnique.fr/perso/sabbah.claude/livres.html.

[Sil85] H. J. Silverstone, “JWKB connection-formula problem revisited via Borel summation”,Phys. Rev. Lett. 55 no. 23, (1985) 2523–2526.

http://arxiv.org/abs/arXiv:1607.06931

http://dx.doi.org/10.1112/plms/s3-55.1.59

http://dx.doi.org/10.1112/plms/s3-55.1.59

http://dx.doi.org/10.1007/BF02395062

http://dx.doi.org/10.1007/BF02395062

http://dx.doi.org/10.1007/s11005-016-0842-x

http://dx.doi.org/10.1007/s11005-016-0842-x

http://dx.doi.org/10.2140/gt.2012.16.1609

http://dx.doi.org/10.2140/gt.2012.16.1609

https://books.google.ca/books?id=pMyEmNBkyBsC

https://books.google.ca/books?id=pMyEmNBkyBsC

http://dx.doi.org/10.1088/1751-8113/47/47/474009

http://dx.doi.org/10.1088/1751-8113/47/47/474009

http://dx.doi.org/10.1007/s00208-004-0528-z

http://dx.doi.org/10.4310/AJM.1998.v2.n4.a11

http://www.cmls.polytechnique.fr/perso/sabbah.claude/livres.html

http://dx.doi.org/10.1103/PhysRevLett.55.2523

[Sim90] C. T. Simpson, “Harmonic bundles on noncompact curves”, J. Amer. Math. Soc. 3 no. 3,(1990) 713–770.

[Sim94] C. T. Simpson, “Moduli of representations of the fundamental group of a smoothprojective variety. I”, Inst. Hautes Etudes Sci. Publ. Math. no. 79, (1994) 47–129.http://www.numdam.org/item?id=PMIHES_1994__79__47_0.

[Sim97] C. Simpson, “The Hodge filtration on nonabelian cohomology”, in Algebraic geometry—Santa Cruz 1995, vol. 62 of Proc. Sympos. Pure Math., pp. 217–281. Amer. Math. Soc.,Providence, RI, 1997.

[Sim10] C. T. Simpson, “Iterated destabilizing modifications for vector bundles with connec-tion”, in Vector bundles and complex geometry, vol. 522 of Contemp. Math., pp. 183–206.Amer. Math. Soc., Providence, RI, 2010.

[Str84] K. Strebel, Quadratic differentials, vol. 5 of Ergebnisse der Mathematik und ihrer Gren-zgebiete (3) [Results in Mathematics and Related Areas (3)]. Springer-Verlag, Berlin,1984.

[Sut14] T. Sutherland, Stability conditions for Seiberg-Witten quivers. PhD thesis, University ofSheffield, 2014.

[Vor83] A. Voros, “The return of the quartic oscillator. The complex WKB method”, Annales del’I.H.P. Physique theorique 39 no. 3, (1983) 211–338. http://eudml.org/doc/76217.

[Was76] W. Wasow, Asymptotic expansions for ordinary differential equations. Robert E. KriegerPublishing Co., Huntington, N.Y., 1976. Reprint of the 1965 edition.

[Zol06] H. Zoladek, The Monodromy Group. Monografie Matematyczne. Birkhauser Basel,2006. https://books.google.ca/books?id=U7q2F5gkYLsC.

http://dx.doi.org/10.1090/S0894-0347-1990-1040197-8

http://dx.doi.org/10.1090/S0894-0347-1990-1040197-8

http://www.numdam.org/item?id=PMIHES_1994__79__47_0

http://dx.doi.org/10.1090/pspum/062.2/1492538

http://dx.doi.org/10.1090/conm/522/10300

http://dx.doi.org/10.1090/conm/522/10300

http://dx.doi.org/10.1007/978-3-662-02414-0

http://eudml.org/doc/76217

https://books.google.ca/books?id=U7q2F5gkYLsC

Documents

Nikita Nikolaev - tspace.library.utoronto.ca · Abstract Abelianisation of Logarithmic Connections Nikita Nikolaev Doctor of Philosophy Graduate Department of Mathematics University