Secondary Products in Supersymmetric Field...

Secondary Products in Supersymmetric Field

Theory

Mathew Bullimore

with D. Ben-Zvi, C. Beem, T. Dimofte, A. Neitzke

Part I : Introduction

Introduction I

I would like to talk about some aspects of TQFTs that arise from

‘topological twisting’ a supersymmetric quantum field theory.

The story is well-known to mathematicians through the formalism of the

Cobordism Hypothesis and derived algebraic geometry.

Our aim was to extract a key structure that emerges from this formalism

and understand it concretely in familiar examples.

This is the structure of ‘higher products’.

Introduction II

An important idea in this talk is ‘topological descent’.

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Starting from a local operator O, we can construct extended operators

O(1), O(2), . . .

I O(p) can be wrapped on cycles in Hp(MD,Z).

I This plays an important role in mathematical applications -

Donaldson theory, Gromov-Witten theory

Introduction III

Here we use descendent operators to define a ‘secondary product’ on

local operators.

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O(D1)2

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I In D-dimensions, the secondary product O1,O2 is defined by

surrounding O1 with the descendent OD−12 wrapping SD−1.

I This defines a kind of ‘Poisson bracket’ of degree 1−D.

I Some examples known to physicists but not systematically explored.

Part II : General Setup

Twisted Superalgebra

I will consider a twisted supersymmetric theory in D-dimensions with

superalgebra

Q2 = 0 [Q,Qµ] = iPµ [Qµ, Qν ] = 0

where Q / Qµ transform as scalar / vector with respect to

Spin(D)′ ⊂ Spin(D)×GR .

I Graded commutator [a, b] := ab− (−1)F (a)F (b)ba where F is Z2

fermion number.

I I assume F lifts to a Z grading with F (Q) = 1 and F (Qµ) = −1

(combination of flavour and R-symmetry).

Topological Operators

Topological operators are annihilated by the scalar supercharge

QO = 0 .

Their correlation functions have two important properties:

I They depend only on Q-cohomology classes,

〈Q(O1)O2 · · · 〉 = 〈Q(O1O2 · · · ) 〉 = 0 .

I They are independent of position,

∂µ〈O1(x)O2(y) · · ·〉 = 〈 ∂µO1(x)O2(y) · · ·〉= 〈QQµO1(x)O2(y) · · ·〉= 〈Q(QµO1(x)O2(y) · · · )〉= 0

Q-cohomology

This motivates introduction of the Q-cohomology

A := Im(Q)/Ker(Q) .

I I denote cohomology classes [O(x)] = [O].

I They are independent of position by same argument as before

∂µ[O(x)] = [∂µO(x)] = [Q(QµO(x))] = 0 .

I The Q-cohomology inherits Z-grading by F ,

A =⊕p∈ZAp .

The Primary Product

In dimension D ≥ 2, there is a unique product

∗ : A⊗A → A

defined by

[O1] ∗ [O2] := [O1(x1)O2(x2)] .

This has the properties

I It is will defined: independent of x1, x2.

I Graded commutative: O1 ∗ O2 = (−1)F1F2O2 ∗ O1

I Associative: O1 ∗ (O2 ∗ O3) = (O1 ∗ O2) ∗ O3

We claim that A also inherits a Poisson structure...

Topological Descent I

The first step is to consider the descent construction:

I Start from a topological operator O(x).

I Define sequence of descendent p-form operators

O(p)(x) :=1

p!Qµ1· · ·Qµp

O(x) dxµ1 ∧ · · · ∧ dxµp

I They obey the descent equations

QOp(x) = dO(p−1)(x)

as a consequence of [Q,Qµ] = iPµ.

Example: QO(1) = Q(QµO) dxµ = Q,QµO dxµ = iPµO dxµ = dO

Topological Descent II

We can integrate the descendent over a p-chain γ

O(γ) :=

∫γ

O(p)(x) .

If γ is closed (∂γ = 0) then O(γ) is topological,

QO(γ) =

∫γ

dO(p−1) =

∫∂γ

O(p−1) = 0 .

I The Q-cohomology class [O(γ)] depends only on the homology class

[γ] ∈ Hp(MD,Z).

I Such classes play an important role in mathematical applications:

Donaldson theory, Gromov-Witten theory.

I Does not product anything new on MD = RD but...

The Secondary Product I

Descendent operators may be used to define a ‘secondary product’

, : A⊗A → A

defined by O1,O2 := [O1(SD−1x )O2(x) ] .

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O1(SD1x )

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I SD−1x is a sphere surrounding the point x.

I Cohomology class independent of radius of SD−1x .

The Secondary Product II

The secondary product has the following properties:

I It has degree 1−D with respect to the Z-grading.

I It has definite symmetry,

O2,O1 = (−1)F1F2+DO1,O2 .

I It is a graded derivation over the primary product,

O1,O2 ∗ O3 = O1,O2 ∗ O3 + (−1)(F1+D−1)F2O2 ∗ O1,O3 .

I It obeys the Jacobi identity.

This endows A with the structure of ‘PD-algebra’.

More Sophisticated Viewpoint

Consider a pair of operators inserted at points x1 6= x2.

The configuration space is homotopically

C2(RD) ∼ SD−1 .

There is a product for each homology class

Hp(SD−1,Z) =

Z if p = 0 → primary product ∗Z if p = D − 1 → secondary product , ∅ otherwise

Part III : Two Dimensions

The B-Twist

The N = (2, 2) supersymmetry on M2 = R2 has four supercharges Q+,

Q−, Q+, Q− obeying

[Q+, Q+] = 2iPz [Q−, Q−] = 2iPz .

I I will consider the combinations

Q := Q+ + Q− Qµ =

(Q−

Q+

)

I They transform as a scalar / vector with respect to the diagonal

subgroup U(1)′ ⊂ U(1)× U(1)A.

I They are known as the B-type supercharges.

Example: Chiral Multiplet

In the B-twist a single chiral multiplet consists of

I Scalar boson φ

I Scalars fermions η, ζ

I 1-form fermion χ

S =

∫dφ ∧ ∗dφ+ ζ ∧ dχ+ χ ∧ ∗dη

We choose the Z-grading by U(1)V .

φ φ η ζ χ

U(1)V 0 0 1 1 −1

The supersymmetry transformations are (Q = Qµdxµ)

Qφ = 0 Qζ = 0 Qη = 0 Qφ = η Qχ = dφ

Qφ = χ Qζ = − ∗ dφ Qη = dφ Qφ = 0 Qχ = 0 .

Primary Product

Topological operators are polynomials in φ, φ, η, ζ modulo Qφ = η, which

are simply polynomials in φ, ζ.

Two geometric interpretations:

I We haveA = H0,•

∂(C,∧•TC)

under the identification

Q ∼ ∂ η ∼ dφ ζ ∼ ∂

∂φ.

I Recalling the Z-grading

φ ζ

U(1)V 0 1

this is functions on the shifted cotangent bundle T ∗[1]C.

Secondary Product I

In dimension D = 2, the secondary bracket has degree −1.

From the Z-grading

φ χ

U(1)V 0 1

the only possible non-vanishing bracket is ζ, φ ∼ 1.

To compute the bracket, we descend once on ζ.

I The first descendent is ζ(1) = Qζ = − ∗ dφ .I It’s exterior derivative is an equation of motion,

dζ(1) = −d ∗ dφ =δS

δφ.

Secondary Product II

Here is the computation of the bracket,

ζ, φ =

∮S1x

ζ(1)φ(x)

=

∫D2

x

dζ(1)φ(x)

=

∫D2

x

(−d ∗ dφ)φ(x)

=

∫D2

x

δS

δφφ(x) = 1 .

I Schouten-Nijenhuis bracket on holomorphic polyvector fields on

X = C (the unique extension of the Lie bracket).

I Poisson structure on shifted cotangent bundle T ∗[1]C.

General Kahler Target

Consider a supersymmetric sigma model to a Kahler target X.

The topological algebra in the B-model is the Dolbeault cohomology of

holomorphic polyvector fields

A =⊕p,q

H0,p

∂(X,∧qTX)

I The Z-grading by U(1)V is p+ q

I Due to the absence of instanton corrections, the primary product

coincides with the wedge product of polyvector fields.

I The secondary product , is Schouten-Nijenhuis bracket.

Part IV : Three Dimensions

3d N = 4 Theories

We have supercharges QAAα transforming in the tri-fundamental

representation of

SU(2)× SU(2)H × SU(2)C .

There are two topological twists:

I H-twist: SU(2)′ ⊂ SU(2)× SU(2)H

I C-twist: SU(2)′ ⊂ SU(2)× SU(2)C

Specification of the supercharges Q, Qµ requires a further choice of

complex structure on the Coulomb / Higgs branch.

Today I will consider the C-twist. [Rozansky-Witten]

Hypermultiplet

After performing the C-twist, a single hypermultiplet consists of

I Complex scalar bosons XI

I Scalar fermions ηI

I 1-form fermions χI

where I = 1, 2 index C2 with holomorphic symplectic form ΩIJ .

S =

∫dXI ∧ ∗dXI + ΩIJχ

I ∧ dχJ + ηId ∗ χI

The supersymmetry transformations are (Q = Qµdxµ)

QXI = 0 QXI = ηI QηI = 0 QχI = dXI

QXI = χIµ QXI = 0 QηI = dXI QχI = ΩIJdXJ .

Primary Product

The topological operators are polynomials in XI , XI ηI modulo the

relation QXI = ηI , which are simply polynomials in XI .

Two geometric interpretations:

I We have A = H0,•(T ∗C) under the identification

ηI ∼ dXI Q ∼ dXI∂XI.

I Introducing the Z-grading

X1 X2 η1, χ1 η2, χ

2

U(1)′H 2 0 1 −1

(a combination of R-symmetry and flavour symmetry) this is

functions on the shifted cotangent bundle T ∗[2]C.

Secondary Product I

In dimension D = 3, the secondary product has degree −2.

The only possible non-vanishing bracket is X1, X2 ∼ 1 or more

invariantly XI , XJ ∼ ΩIJ .

Let us compute the descendents of XI :

I The first descendent is (XI)(1) = QXI = χIµ.

I The second descendent is (XI)(2) = QχI = ΩIJ ∗ dXJ .

I Finally, the exterior derivative is an equation of motion,

d(XI)(2) = ΩIJd ∗ dXJ = ΩIJδS

δXJ.

Secondary Product II

We now compute the secondary product as follows

XI , XJ =

∫S2x

(XI)(2)XJ(x)

=

∫D3

x

d(XI)(2)XJ(x)

= ΩIK∫D3

x

δS

δXKXJ(x)

= ΩIJ

I This is tantamount to the holomorphic symplectic structure on the

Higgs branch MH = T ∗C.

I Alternatively, keeping track of the Z-grading, it is the Poisson

structure on the shifted cotangent bundle T ∗[2]C.

Supersymmetric Gauge Theory

A 3d N = 4 gauge theory has two important moduli spaces of vacua

1. Higgs branch MH

2. Coulomb branch MC

which are both holomorphic symplectic varieties.

I C-twist : ( A , , ) coincide with the coordinate ring of MH and

its holomorphic Poisson bracket.

I H-twist : ( A , , ) coincide with the coordinate ring of MC and

its holomorphic Poisson bracket.

The physical construction of the holomorphic symplectic structure is new!

Part V : Conclusions

Topological Defects

The formalism presented here can be generalised to ‘higher products’ of

extended topological operators.

This is a shadow of full machinery of Cobordism Hypothesis.

One example: line operators form a braided tensor category.

S1

L M L M

RLM

The braiding is an example of a ‘secondary product’ of line operators.

Omissions

I Examples of higher products involving extended operators.

I Ω-deformation and deformation quantisation of the secondary

bracket , .

I Topological twists in four dimensions and connections to geometric

Langlands.

Future Directions

I Systematic exploration of higher products of extended operators.

I Construction of ED−k-monoidal structure of categories of

k-dimensional operators.

I The role this plays in ‘generalised global symmetries’.

[Gaiotto-Kapustin-Seiberg-Willet]

I Higher products on ‘holomorphically twisted’ supersymmetric field

theories. [Costello]