Line defects, UV-IR map, and exact WKB

Fei Yan

Rutgers University

Donaldson-Thomas invariants and Resurgence,

Simons Collaboration meeting, Jan 11 - Jan 15, 2021

Jan 13th, 2021

Outlook

Class S theories Hitchin systems

Line defects ???

Based on review of work by Gaiotto-Moore-Neitzke, as well assome recent developments.

4d N=2 theories of class S

Class S theories T rg,C s are 4d N=2 supersymmetric theoriesoriginating from twisted compactification of a 6d p2, 0q theory oftype g (g P tA,D,Eu) on a Riemann surface C with appropriatedecorations (punctures or twisted lines), in the zero area limit of C .[Gaiotto],[Gaiotto-Moore-Neitzke]

T rg,C s has a subspace of vacua B called the Coulomb branch,where the low energy effective theory is Up1qr gauge theory.

B Ărà

k“1

H0

˜

C ,KbdkC

˜

ÿ

i

ppiqdkzi

¸¸

[Gaiotto],[Gaiotto-Moore-Neitzke],[Chacaltana-Distler-Tachikawa],[Chacaltana-Distler-Trimm-Zhu] ...

A point in B corresponds to a branched covering rC Ñ C , whererC Ă T ˚C is the Seiberg-Witten curve.

4d N=2 theories of class S

Class S theories T rg,C s are 4d N=2 supersymmetric theoriesoriginating from twisted compactification of a 6d p2, 0q theory oftype g (g P tA,D,Eu) on a Riemann surface C with appropriatedecorations (punctures or twisted lines), in the zero area limit of C .[Gaiotto],[Gaiotto-Moore-Neitzke]

T rg,C s has a subspace of vacua B called the Coulomb branch,where the low energy effective theory is Up1qr gauge theory.

B Ărà

k“1

H0

˜

C ,KbdkC

˜

ÿ

i

ppiqdkzi

¸¸

[Gaiotto],[Gaiotto-Moore-Neitzke],[Chacaltana-Distler-Tachikawa],[Chacaltana-Distler-Trimm-Zhu] ...

A point in B corresponds to a branched covering rC Ñ C , whererC Ă T ˚C is the Seiberg-Witten curve.

Relation to Hitchin systems

Further compactifying T rg,C s on S1R , the low energy effective

theory is a 3d N “ 4 sigma model with target space MHpG ,C q.

Starting from 6d and changing the order of compactification onC ˆ S1

R [Gaiotto-Moore-Neitzke], MHpG ,C q is identified with themoduli space of solutions to Hitchin’s equations:

FA ` R2“

Φ, Φ‰

“ 0,

BAΦ “ 0, BAΦ “ 0.

B ` A is a G -connection in a top. trivial G -bundle V Ñ C ,Φ P Ω1,0pEndV q is the Higgs field.

pA,Φq have specified boundary conditions at punctures of C .(regular: simple pole, irregular: higher order pole)

Today we take g “ glpNq or slpNq, G “ UpNq or SUpNq.

Relation to Hitchin systems

Further compactifying T rg,C s on S1R , the low energy effective

theory is a 3d N “ 4 sigma model with target space MHpG ,C q.

Starting from 6d and changing the order of compactification onC ˆ S1

R [Gaiotto-Moore-Neitzke], MHpG ,C q is identified with themoduli space of solutions to Hitchin’s equations:

FA ` R2“

Φ, Φ‰

“ 0,

BAΦ “ 0, BAΦ “ 0.

B ` A is a G -connection in a top. trivial G -bundle V Ñ C ,Φ P Ω1,0pEndV q is the Higgs field.

pA,Φq have specified boundary conditions at punctures of C .(regular: simple pole, irregular: higher order pole)

Today we take g “ glpNq or slpNq, G “ UpNq or SUpNq.

Relation to Hitchin systems

MHpG ,C q is hyperkahler, has a CP1-worth of complex structuresJζ . Different Jζ expose different features of MHpG ,C q:[Hitchin], [Simpson], [Biquard-Boalch], [Gaiotto-Moore-Neitzke]...

§ ζ P Cˆ: Hitchin’s equations indicate B `A is flat, with

A :“R

ζΦ` A` RζΦ

pMH , Jζq diff. to a moduli space of flat GC-connections on C .

§ ζ “ 0: pMH , J0q diff. to moduli space of Higgs bundlesMHiggs, which is a complex integrable system: MHiggs Ñ Bwith generic fiber being compact tori.

The Seiberg-Witten curve rC identified with spectral curve:

rC “ tpz P C , λ P T ˚z C q : Det pΦpzq ´ λq “ 0u Ă T ˚C

Relation to Hitchin systems

MHpG ,C q is hyperkahler, has a CP1-worth of complex structuresJζ . Different Jζ expose different features of MHpG ,C q:[Hitchin], [Simpson], [Biquard-Boalch], [Gaiotto-Moore-Neitzke]...

§ ζ P Cˆ: Hitchin’s equations indicate B `A is flat, with

A :“R

ζΦ` A` RζΦ

pMH , Jζq diff. to a moduli space of flat GC-connections on C .

§ ζ “ 0: pMH , J0q diff. to moduli space of Higgs bundlesMHiggs, which is a complex integrable system: MHiggs Ñ Bwith generic fiber being compact tori.

The Seiberg-Witten curve rC identified with spectral curve:

rC “ tpz P C , λ P T ˚z C q : Det pΦpzq ´ λq “ 0u Ă T ˚C

Line defects in class S theories

T rg,C s admits families of line defects Lpζq extending alongRt-direction, where ζ P Cˆ parametrizes preserved supercharges.[Kapustin],[Kapustin-Saulina],[Drukker-Morrison-Okuda],[Drukker-Gaiotto-Gomis],[Drukker-Gomis-Okuda-Teschner]

[Gaiotto-Moore-Neitzke],[Cordova-Neitzke],[Aharony-Seiberg-Tachikawa],[Moore-Royston-van den Bleeken]...

Lpζ, p,Rq depends on path p on C , carrying representation R of g.

§ g “ A1, C only has regular punctures:p is a non-self-intersecting closed curve on C .

§ C contains irregular punctures:p corresponds to integral laminations [Fock-Goncharov],[Gaiotto-Moore-Neitzke],collection of paths either closed or open with ends on markedpoints corres. to Stokes directions at irregular punctures.

§ In general p could contain junctions, where paths carryingdifferent Ri meet, associated with certain g-invariant tensor.[Sikora],[Le],[Xie],[Saulina],[Coman-Gabella-Teschner],[Tachikawa-Watanabe],[Gabella]...

Line defects in class S theories

T rg,C s admits families of line defects Lpζq extending alongRt-direction, where ζ P Cˆ parametrizes preserved supercharges.[Kapustin],[Kapustin-Saulina],[Drukker-Morrison-Okuda],[Drukker-Gaiotto-Gomis],[Drukker-Gomis-Okuda-Teschner]

[Gaiotto-Moore-Neitzke],[Cordova-Neitzke],[Aharony-Seiberg-Tachikawa],[Moore-Royston-van den Bleeken]...

Lpζ, p,Rq depends on path p on C , carrying representation R of g.

§ g “ A1, C only has regular punctures:p is a non-self-intersecting closed curve on C .

§ C contains irregular punctures:p corresponds to integral laminations [Fock-Goncharov],[Gaiotto-Moore-Neitzke],collection of paths either closed or open with ends on markedpoints corres. to Stokes directions at irregular punctures.

§ In general p could contain junctions, where paths carryingdifferent Ri meet, associated with certain g-invariant tensor.[Sikora],[Le],[Xie],[Saulina],[Coman-Gabella-Teschner],[Tachikawa-Watanabe],[Gabella]...

Line defects in class S theories

Upon circle compactification, the vacuum expectation values ofLpζq wrapping S1

R are Jζ-holomorphic functions on MflatpGC,C q.[Gaiotto-Moore-Neitzke]

§ g “ A1, C has only regular punctures:

xLpζ, p,Rqy “ TrRHolp

ˆ

RΦ

ζ` A` RζΦ

˙

“ TrRHolpApζq.

The dependence on p is only through its homotopy class.

§ In general, compute parallel transport of Apζq along paths,contract together via g-invariant tensors.

Line defects in class S theories

Upon circle compactification, the vacuum expectation values ofLpζq wrapping S1

R are Jζ-holomorphic functions on MflatpGC,C q.[Gaiotto-Moore-Neitzke]

§ g “ A1, C has only regular punctures:

xLpζ, p,Rqy “ TrRHolp

ˆ

RΦ

ζ` A` RζΦ

˙

“ TrRHolpApζq.

The dependence on p is only through its homotopy class.

§ In general, compute parallel transport of Apζq along paths,contract together via g-invariant tensors.

The UV-IR map for line defects

A useful way to study Lpζq in class S theories, is deforming to apoint u on the Coulomb branch B and follow the defect into IR.

The IR limit of Lpζq is a superposition of supersymmetric linedefects in the abelian theory, with integer coefficients in thissuperposition given by framed BPS index ΩpLpζq, γ, uq.[Gaiotto-Moore-Neitzke],[Cordova-Neitzke],[Cirafici-Del Zotto],[Coman-Gabella-Teschner],

[Moore-Royston-van den Bleeken],[Ito-Okuda-Taki],[Galakhov-Longhi-Moore],. . .

The UV-IR map for line defects:

Lpζq ùÿ

γ

ΩpLpζq, γ, uqXγpζq

Xγpζq represent IR Wilson-’t Hooft lines with charge γ.

The UV-IR map for line defects

A useful way to study Lpζq in class S theories, is deforming to apoint u on the Coulomb branch B and follow the defect into IR.

The IR limit of Lpζq is a superposition of supersymmetric linedefects in the abelian theory, with integer coefficients in thissuperposition given by framed BPS index ΩpLpζq, γ, uq.[Gaiotto-Moore-Neitzke],[Cordova-Neitzke],[Cirafici-Del Zotto],[Coman-Gabella-Teschner],

[Moore-Royston-van den Bleeken],[Ito-Okuda-Taki],[Galakhov-Longhi-Moore],. . .

The UV-IR map for line defects:

Lpζq ùÿ

γ

ΩpLpζq, γ, uqXγpζq

Xγpζq represent IR Wilson-’t Hooft lines with charge γ.

The IR line defects

Recall that, a point u on the Coulomb branch B corresponds to abranched covering rC Ñ C (spectral curve/Seiberg-Witten curve):

Geometrically the IR line defect Xγpζq correspond to loops p Ă rC

in class γ P H1p rC ,Zq.

The IR line defects

Upon circle compactification, the VEV Xγpζq of Xγpζq wrappingS1R are local Darboux coordinates on MflatpGC,C q:

Fock-Goncharov, complexified Fenchel-Nielsen, or more generalspectral coordinates.[Fock-Goncharov],[Fenchel-Nielsen],[Gaiotto-Moore-Neitzke],[Nekrasov-Rosly-Shatashvili],

[Hollands-Neitzke],[Hollands-Kidwai],[Allegretti],[Nikolaev],[Jeong-Nekrasov],[Coman-Longhi-Teschner] ...

c.f. Bridgeland’s talk and Coman’s talk

§ The holomorphic Poisson brackets are given by

tXγ ,Xγ1u “ xγ, γ1yXγXγ1

§ Xγ has nice asymptotic behavior as ζ Ñ 0 and ζ Ñ8;has discontinuities across “BPS rays” controlled byKontsevich-Soibelman symplectomorphisms.Xγ are solutions to a certain Riemann-Hilbert problem.[Gaiotto-Moore-Neitzke],[Gaiotto],[Bridgeland],[Barbieri],[Bridgeland-Barbieri-Stoppa]...

The IR line defects

Upon circle compactification, the VEV Xγpζq of Xγpζq wrappingS1R are local Darboux coordinates on MflatpGC,C q:

Fock-Goncharov, complexified Fenchel-Nielsen, or more generalspectral coordinates.[Fock-Goncharov],[Fenchel-Nielsen],[Gaiotto-Moore-Neitzke],[Nekrasov-Rosly-Shatashvili],

[Hollands-Neitzke],[Hollands-Kidwai],[Allegretti],[Nikolaev],[Jeong-Nekrasov],[Coman-Longhi-Teschner] ...

c.f. Bridgeland’s talk and Coman’s talk

§ The holomorphic Poisson brackets are given by

tXγ ,Xγ1u “ xγ, γ1yXγXγ1

§ Xγ has nice asymptotic behavior as ζ Ñ 0 and ζ Ñ8;has discontinuities across “BPS rays” controlled byKontsevich-Soibelman symplectomorphisms.Xγ are solutions to a certain Riemann-Hilbert problem.[Gaiotto-Moore-Neitzke],[Gaiotto],[Bridgeland],[Barbieri],[Bridgeland-Barbieri-Stoppa]...

The UV-IR map as the trace map

§ xLpζqy are Jζ-holomorphic trace functions on MflatpGC,C q.

§ Xγpζq :“ xXγpζqy are Darboux-coordinates on MflatpGC,C q.

§ The UV-IR map then implies the trace map:

TrRHolpApζq “ÿ

γ

ΩpLpζq, γqXγpζq

Apζq: flat GC-connection on C , p: path on C .

§ Alternatively we can understand Xγpζq as:

Xγpζq “ Holγ∇abpζq, γ P H1p rC ,Zq.

The UV-IR map is interpreted as a nonabelianization map.[Gaiotto-Moore-Neitzke],[Hollands-Neitzke]

The UV-IR map as the trace map

§ xLpζqy are Jζ-holomorphic trace functions on MflatpGC,C q.

§ Xγpζq :“ xXγpζqy are Darboux-coordinates on MflatpGC,C q.

§ The UV-IR map then implies the trace map:

TrRHolpApζq “ÿ

γ

ΩpLpζq, γqXγpζq

Apζq: flat GC-connection on C , p: path on C .

§ Alternatively we can understand Xγpζq as:

Xγpζq “ Holγ∇abpζq, γ P H1p rC ,Zq.

The UV-IR map is interpreted as a nonabelianization map.[Gaiotto-Moore-Neitzke],[Hollands-Neitzke]

Line defects OPE

The algebra structure on space of Jζ-holomorphic functionscorresponds to line defects operator products (OPE):

xL1pζqL2pζqy “ xL1pζqyxL2pζqy

This algebra structure admits a quantization via skein algebras.[Reshetikhin-Turaev],[Turaev],[Witten],[Alday-Gaiotto-Gukov-Tachikawa-Verlinde],[Gaiotto-Moore-Neitzke],

[Drukker-Gomis-Okuda-Teschner],[Tachikawa-Watanabe],[Coman-Gabella-Teschner],[Gabella]...

Turning on Ω-deformation on a R2-plane, susy line defects sitalong a fixed axis in R3 ù non-commutative associative OPE ˚[Nekrasov-Shatashvili],[Gaiotto-Moore-Neitzke],[Ito-Okuda-Taki],[Yagi],[Oh-Yagi],...

Line defects OPE

The algebra structure on space of Jζ-holomorphic functionscorresponds to line defects operator products (OPE):

xL1pζqL2pζqy “ xL1pζqyxL2pζqy

This algebra structure admits a quantization via skein algebras.[Reshetikhin-Turaev],[Turaev],[Witten],[Alday-Gaiotto-Gukov-Tachikawa-Verlinde],[Gaiotto-Moore-Neitzke],

[Drukker-Gomis-Okuda-Teschner],[Tachikawa-Watanabe],[Coman-Gabella-Teschner],[Gabella]...

Turning on Ω-deformation on a R2-plane, susy line defects sitalong a fixed axis in R3 ù non-commutative associative OPE ˚[Nekrasov-Shatashvili],[Gaiotto-Moore-Neitzke],[Ito-Okuda-Taki],[Yagi],[Oh-Yagi],...

Line defects OPE

§ In the UV, non-commutative line defects OPE is complicated.

§ In the IR, the OPE is given by quantum torus algebra:

Xγ1pζq ˚ Xγ2pζq “ p´qqxγ1,γ2yXγ1`γ2pζq.

x, y is the Dirac-Schwinger-Zwanziger pairing, identified withintersection pairing in H1p rC ,Zq.In particular,

Xγ1pζq ˚ Xγ2pζq “ p´qq2xγ1,γ2yXγ2pζq ˚ Xγ1pζq

quantizes the Poisson algebra of coordinate functions Xγ .

§ This hints existence of a quantum trace map, embedding theUV skein algebra into a quantum torus algebra.

The UV skein algebra

For example, taking g “ glp2q, the space of line defects equippedwith OPE is described by the glp2q HOMFLY skein algebra ofM “ C ˆ Rh, defined as the space of formal Zrq˘1s-linearcombinations of framed oriented links (up to isotopy) in M,modulo the following relations: (N=2 in the figure)

The algebra structure is defined by “stacking” links along theRh-direction.

The IR skein algebra

The IR line defects correspond to framed oriented links inrM “ rC ˆ R. The OPE algebra is (twisted) glp1q skein algebra ofrM, defined as the space of formal Zrq˘1s-linear combinations of

framed oriented links (up to isotopy) in rM, modulo the followingrelations:

This skein algebra is isomorphic to the quantum torus.

The quantum trace map

The quantum trace map F is a homomorphism from the UV skeinalgebra to the IR quantum torus algebra, physically it represents aq-deformed UV-IR map:

Lpζq ùÿ

γ

ΩpLpζq, γ, u, qqXγpζq

ΩpLpζq, γ, u, qq :“ TrHL,γ,up´qq2J3q2I3e´βtQ,Q

`u P Zrq, q´1s

[Gaiotto-Moore-Neitzke],[Fock-Goncharov],[Bonahon-Wong],[Cordova-Neitzke],[Cirafici-Del Zotto],

[Coman-Gabella-Teschner],[Allegretti],[Gabella],[Ito-Okuda-Taki],[Galakhov-Longhi-Moore],[Neitzke-Y],

[Cho-Kim-Kim-Oh],[Le],[Kim-Son],[Korinman-Quesney],[Goncharov-Shen],[Douglas-Sun]...

§ When M “ R3, F computes a familiar link invariant:

F pLq “ qNwpLqPHOMFLYpL, a “ qN , z “ q ´ q´1q,

where wpLq is the self-linking number of L Ă M.It also works if M “ C ˆ R and L is contained in a 3-ballinside M. [Bonahon-Wong],[Neitzke-Y],[Douglas-Sun]

The quantum trace map

The quantum trace map F is a homomorphism from the UV skeinalgebra to the IR quantum torus algebra, physically it represents aq-deformed UV-IR map:

Lpζq ùÿ

γ

ΩpLpζq, γ, u, qqXγpζq

ΩpLpζq, γ, u, qq :“ TrHL,γ,up´qq2J3q2I3e´βtQ,Q

`u P Zrq, q´1s

[Gaiotto-Moore-Neitzke],[Fock-Goncharov],[Bonahon-Wong],[Cordova-Neitzke],[Cirafici-Del Zotto],

[Coman-Gabella-Teschner],[Allegretti],[Gabella],[Ito-Okuda-Taki],[Galakhov-Longhi-Moore],[Neitzke-Y],

[Cho-Kim-Kim-Oh],[Le],[Kim-Son],[Korinman-Quesney],[Goncharov-Shen],[Douglas-Sun]...

§ When M “ R3, F computes a familiar link invariant:

F pLq “ qNwpLqPHOMFLYpL, a “ qN , z “ q ´ q´1q,

where wpLq is the self-linking number of L Ă M.It also works if M “ C ˆ R and L is contained in a 3-ballinside M. [Bonahon-Wong],[Neitzke-Y],[Douglas-Sun]

The quantum trace map: an example

Example: g “ slp2q, C is a four-punctured sphere.

rC “ tλ : λ2 ` φ2 “ 0u Ă T ˚C , φ2 “ ´z4 ` 2z2 ´ 1

2pz4 ´ 1q2dz2.

2

1

1

2

2

2

1

1

1

1

2

2

2

2

1

11

1

1 2

2

2

2

1

0

1

2

4

7

5

3

6

2

Properties of Xγpζq

The VEVs of IR line defects Xγpζq have distinguished analyticproperties: c.f. Bridgeland’s talks

§ Asymptotics: As ζ Ñ 0 along a ray in Cˆ:

Xγpζq „ exp

ˆ

R

ζZγ

˙

,

where Zγ “ű

γ λ is central charge/classical period. c.f. Smith’s talk

§ Jumps: Given a ray ` Ă Cˆ with phase θ0, then:

limargpζqÑθ`0

Xγpζq “ Sp`q limargpζqÑθ´0

Xγpζq

Sp`q consists of Kontsevich-Soibelman symplectomorphisms,determined by the Donaldson-Thomas invariants Ωpγq.

§ Reality:Xγpζq “ X´γ

`

´1ζ˘

.

The Riemann-Hilbert problem

Xγpζq are conjectured to solve a Riemann-Hilbert (RH) problem inζ-plane, formulated by Gaiotto-Moore-Neitzke; the solutions aregiven by certain TBA-like integral equations. This RH problem isrelated to the RH problem considered by Bridgeland via theconformal limit [Gaiotto].

Barbieri-Bridgeland-Stoppa formulated a quantized RH problemand constructed solutions in certain example.

Q: Could we understand/generalize the construction from physics?

The Riemann-Hilbert problem

Xγpζq are conjectured to solve a Riemann-Hilbert (RH) problem inζ-plane, formulated by Gaiotto-Moore-Neitzke; the solutions aregiven by certain TBA-like integral equations. This RH problem isrelated to the RH problem considered by Bridgeland via theconformal limit [Gaiotto].

Barbieri-Bridgeland-Stoppa formulated a quantized RH problemand constructed solutions in certain example.

Q: Could we understand/generalize the construction from physics?

The conformal limit

Recall the Hitchin system depends on R as the radius of thecompactification circle. The conformal limit is given by [Gaiotto]

R Ñ 0, ζ Ñ 0, ~ “ ζR fixed

The Hitchin section becomes the variety of opers L~.[Dumitrescu-Fredrickson-Kydonakis-Mazzeo-Mulase-Neitzke]

Example: Take GC “ SLp2,Cq, the family of opers is determinedby a quadratic differential φ2 on C , locally written as:

~2B2z ` P2pzq,

where φ2 “ P2pzqdz2.

The spectral coordinates Xγp~q are important tools in exact WKBanalysis for opers, due to the geometric reformulation of exactWKB via abelianization.[Gaiotto-Moore-Neitzke],[Hollands-Neitzke],[Hollands-Kidwai],[Allegretti],[Nikolaev]...

The Voros symbols Xγp~q and quantum periods

WKB ansatz for the Schrodinger equation:

ψpzq “ exp

ˆ

1

~

ż z

z0

λpz 1qdz 1˙

, where λ2 ` P2 ` ~Bzλ “ 0.

The quantum periods:

Πγp~q :“

¿

γ

λformalp~qdz , γ P H1p rC ,Zq, λformali p~q “

8ÿ

n“0

λpnqi ~n

A natural way to resum quantum periods is Borel-resummation.c.f. Pym’s and Marino’s talk

Resurgent properties of quantum periods are controlled by 4d BPSspectrum. [Ito-Marino-Shu],[Grassi-Gu-Marino],[Ito-Shu]...

Relation of spectral coordinates Xγp~q to quantum periods:

§ Xγp~q produces Borel-resummed quantum periods.[Koike-Schafke],[Nikolaev],[Allegretti],...

For higher order opers, this remains to be a conjecture.

Thank You and Stay Safe!