Painlevรฉ/Gauge theory correspondence on the torus

Fabrizio Del Monte, SISSA 1901.10497, 19xx.xxxxx Work in collaboration with G. Bonelli, P. Gavrylenko, A. Tanzini Contents

CFT2 approach to Mathematical Physical Isomonodromic Conclusions and Formulation of motivation deformations on Outlook the problem the torus Class S theories and Hitchin Systems I

6d 5d 4d 3d

3d NLSM with 5ํ‘‘ ํ’ฉ = 2 ํถ๐‘”,ํ‘›,ํœƒ target Hitchin ํ‘†ํ‘Œํ‘€(ํบ) ํ‘†1 moduli space ํ’ฉ = (0,2) Class 1 3d NLSM with ํ’ฎ(ํ”ค) ํ‘† ํถ ํ’ฎ(ํ”ค, ํถ ) target SW ๐‘”,ํ‘›,ํœƒ ๐‘”,ํ‘›,ํœƒ moduli space Class S theories and Hitchin system

โ€ข Data for the 4d theory: ํถ๐‘”,ํ‘›,ํœƒ: ํ‘”, ํ‘› determines the gauge and matter content; ํœƒํ‘˜, ํœŽ determine masses of hypermultiplets and vector vev

ํœ—ิฆ3 ํœ—ิฆ1

ํœ—ิฆ2 ํœ—ิฆ4 Weak coupling regime Degeneration limit of RS

โ€ข Hitchin system: Lax pair ํฟ, ํ‘€ meromorphic matrix-valued differentials on ํถ๐‘”,ํ‘› ํ‘˜ โ€ข ํ‘กํ‘Ÿ ํฟ = ํ‘ขํ‘˜ Hamiltonians of integrable system/Coulomb branch parameters; double pole 2 coefficients of ํ‘กํ‘Ÿ ํฟ are ํœƒํ‘˜โ€™s โ€ข ฮฃ: det(ํฟ โˆ’ ํœ†ํผ) is the SW curve Isomonodromic deformations and class S theories

Complex structure moduli of ํถ๐‘”,ํ‘›

Gauge couplings (or dynamical energy scale in the asymptotically free cases) in weakly coupled frames โ€ข Deforming this picture by varying these moduli does not change the physical theory: the resulting deformation equations are RG equations (asymptotically free) or conformal manifold equations (CFTs) โ€ข These deformations are well-known from the point of view of integrable systems: they are isomonodromic deformations of the Hitchin system โ€ข These flows are Hamiltonian, the generating function for these Hamiltonians is ํท ๐‘–ํ‘›ํœ‚ called the tau function ํ’ฏ = ํ‘ = ฯƒํ‘› ํ‘ ํ‘Ž + ํ‘› ํ‘’ of the corresponding gauge theory in the self-dual Omega background Topological Strings and Hitchin Systems

TS โ€ข Open CY3 X Mirror curve ฮฃํ‘‹ (nontrivial part of the mirror CY3) ํ‘‡ํ‘† โ€ข 4d limit of ฮฃํ‘‹ is the spectral curve ฮฃX: det ํฟ โˆ’ ํœ†ํผ = 0. โ€ข The isomonodromic deformations of the Hitchin system turn out to be closely related to the quantization of the spectral curve. โ€ข The differential equations are exactly satisfied by the dual gauge theory partition function in self-dual Omega background (unrefined TS) [GIL2013,ILT2015,GM2016, BLMST2017] โ€ข If one considers the problem before the 4d limit, the โ€œdeformationโ€ equations are really difference equations (q-Painlevรฉ) satisfied by the TS partition function [BGT2018,BGM2018] โ€ข The tau function can be written as a Fredholm [Z1994,GL2018,CGL2018]: nonperturbative definition of TS! Isomonodromic deformations and quantum mirror curves

โ€ข BGT ๏ƒŸโ†’ GHM: inverse of quantum mirror curve is a operator (see A. Grassiโ€™s talk) โ€ข One can show its Fredholm determinant in the 4d limit is tau function of an isomonodromic problem (pure gauge, 4d and 5d) โ€ข By other constructions ([BLMST]) one knows that this tau function is the NO dual partition function of a gauge theory. The Fredholm determinant gives a nonperturbative representation of the TS partition function. โ€ข CPT (see J. Teschnerโ€™s talk): โ€ข The free fermion representation of the partition function provides the bridge between TS and integrable hierarchies, again the Fredholm determinant provides nonperturbative definition โ€ข Follows the approach of [ADKMV2005,DHSV2008]: free fermion conformal blocks correspond to B-model amplitudes with possible insertion of B-branes. โ€ข The two approaches are complementary, in the sense that BGT probes the magnetic phase of the 4d gauge theory, while CPT the electric phase: two different 4d limits of Topological Strings. Goals

โ€ข Extend this correspondence to the (differential, 4d) case of g=1 with n regular punctures โ€ข 4d gauge theory: circular quiver gauge theories, with matter in the adjoint representation of the gauge (6d theory little strings, see S. Hoheneggerโ€™s talk) โ€ข : explicit solution of isomonodromic problems on higher genus Riemann surfaces is still an open problem Results I

โ€ข On the torus, there is an extra factor between tau function and NO dual partition function/free fermion chiral correlator:<

ํท ํ’ฏํ‘ ํ‘โ„Žํ‘’ํ‘Ÿํ‘’ = ํ‘ โ€ข One puncture case:

ํท ํท ํท ํ‘ ํ‘ ํ‘ ฮค ํ’ฏ = = 0 = 1 2 2 ํœƒ 2ํ‘„ ํœƒ 2ํ‘„ ํœƒ1 ํ‘„ 3 2 ํœ‚ ํœ 2 ํœ‚ ํœ ํœ‚ ํœ

ํ‘‘2ํ‘„ = 2ํœ‹ํ‘–ํ‘š 2โ„˜โ€ฒ 2ํ‘„ ํœ (Maninโ€™s PVI special case) ํ‘‘ํœ2 Results I

โ€ข On the torus, there is an extra factor between tau function and NO dual partition function/free fermion chiral correlator: ํท ํ‘ ํ‘ ํœƒ1 ํ‘„ํ‘– ํ’ฏ = ํ‘ํ‘กํ‘ค๐‘–ํ‘ ํ‘ก = ฯ‚๐‘–=1 ํ‘ํ‘กํ‘คํ‘–ํ‘ ํ‘ก ํ‘„ ํœ‚ ํœ โ€ข One puncture case:

ํท ํท ํท ํ‘ ํ‘ ํ‘ ฮค ํ’ฏ = = 0 = 1 2 2 ํœƒ 2ํ‘„ ํœƒ 2ํ‘„ ํœƒ1 ํ‘„ 3 2 ํœ‚ ํœ 2 ํœ‚ ํœ ํœ‚ ํœ

ํ‘‘2ํ‘„ = 2ํœ‹ํ‘–ํ‘š 2โ„˜โ€ฒ 2ํ‘„ ํœ (Maninโ€™s PVI special case) ํ‘‘ํœ2 Results II

โ€ข The last equation gives us an exact relation between the Painlevรฉ transcendant and the two Fourier transforms (sectors with different fermion charges), that gives an exact relation between the UV and IR coupling of the gauge theory

ํท ํ‘0 ํœƒ3 2ํ‘„ ํœ 2ํœ ํท = ํ‘1ฮค2 ํœƒ2(2ํ‘„(ํœ)|2ํœ)

2 2 2 2 2 2 8 ํœƒ2 2ํœํ‘†ํ‘Š ํœƒ3 2ํœํ‘†ํ‘Š ํœƒ2 2ํœ ํœƒ3 2ํœ ํœ‹ ํœ‡ ํœƒ4 2ํœ 2 + 2 = 2 + 2 โˆ’ 2 2 2 ํœƒ3 2ํœํ‘†ํ‘Š ํœƒ2 2ํœํ‘†ํ‘Š ํœƒ3 2ํœ ํœƒ2 2ํœ ํ‘ข โˆ’ 4ํœ‹ํ‘–ํœ‡ ํœ•ํœ log ํœƒ2 ํœ ํœƒ2 2ํœ ํœƒ3 2ํœ

โ€ข Fredholm determinant 2 representation of the (dual) ํ‘ํท = ํ‘ ํ‘Ž, ํ‘š, ํ‘Ž ํ‘žํ‘Ž det(1 + ํพ) partition function: CFT approach to Mathematical Physical Isomonodromic Conclusions and Formulation of motivation deformations on Outlook the problem the torus Riemann Hilbert Problem and linear system on the sphere

Riemann Hilbert Problem Linear system

โˆ’ํœ—ํ‘˜ ํ‘Œ ํ‘ง~ํ‘งํ‘˜ = ํบํ‘˜ ํ‘ง โˆ’ ํ‘งํ‘˜ ํถํ‘˜ ํœ•ํ‘งํ‘Œ ํ‘ง = ํฟ ํ‘ง ํ‘Œ ํ‘ง ํ‘› เตž det ํ‘Œ ํ‘ง0 โ‰  0 ํดํ‘˜ ํ‘Œ ํ‘ง0 = ํ‘–ํ‘‘ํ‘ ํฟ ํ‘ง = เทŽ ํ‘ง โˆ’ ํ‘งํ‘˜ โ€ข ํœ—ิฆํ‘˜ are the local monodromy exponents ํ‘˜=1 ํ‘Œ ํ›พํ‘˜ โ‹… ํ‘ง = ํ‘Œ ํ‘ง ํ‘€ํ‘˜ โ€ข Kernel: ํ‘Œ ํ‘ง0 = ํ‘–ํ‘‘ํ‘ ํ‘Œโˆ’1 ํ‘ง ํ‘Œ ํ‘ง 0 ํด = ํถ ํœฝ ํถโˆ’1 ํพ ํ‘ง, ํ‘ง0 = ํ‘˜ ํ‘˜ ํ’Œ ํ‘˜ ํ‘ง โˆ’ ํ‘ง0 2ํœ‹๐‘–ํœฝํ’Œ โˆ’1 ํ‘€ํพ = ํถํ‘˜ํ‘’ ํถํ‘˜

Isomonodromic deformations: Deformations of the above linear systems that preserve the conjugacy classes of the monodromies (moving the points ํ‘งํ‘˜) 1 1 2 Isomonodromic tau function: ํœ•ํ‘ง log ํ’ฏ = ํปํ‘˜, ํปํ‘˜ = ืฏ ํ‘กํ‘Ÿํฟ ํ‘‘ํ‘ง ํ‘˜ 2ํœ‹๐‘– ํ›พํ‘˜ 2 Linear Systems on the Torus

On the torus there cannot be a โ€ข Take ํฟ(ํ‘ง) to have ํ‘ additional singularities with residues of rank 1 function with just one simple pole, [Krichever 2002]; so one cannot directly generalize ํฟ ํ‘ง . โ€ข Consider ํฟ(ํ‘ง) to be a section of a nontrivial bundle on the torus There is more than one possibility: [Korotkin, Takasaki 1997-2003].

2ํœ‹ํ‘–ํ‘˜ Flat ํ‘†ํฟ(ํ‘, โ„‚) bundles on the torus can be classified by an element ํ‘’ ํ‘ of their center โ„คํ‘, i.e. by an integer ํ‘˜ = 0, โ€ฆ , ํ‘ โˆ’ 1 through their transition functions [Levin et al. 2014]

2ํœ‹๐‘–ํ‘˜ โˆ’1 โˆ’1 โˆ’1 โˆ’1 ํ‘ ํฟ ํ‘ง + 1 = ํ‘‡ํดํฟ ํ‘ง ํ‘‡ํด , ํฟ ํ‘ง + ํœ = ํ‘‡ํตํฟ ํ‘ง ํ‘‡ํต , ํ‘‡ํดํ‘‡ํตํ‘‡ํด ํ‘‡ํต = ํ‘’

Both twists and monodromies ํœ•ํ‘งํ‘Œ = ํฟํ‘Œ ํ‘Œ โ†’ ํ‘‡ํ‘Œํ‘€ Linear System on the Torus

Bundles with ํ‘˜ = 0

2ํœ‹๐‘–ํ‘„1 2ํœ‹๐‘–ํ‘„ํ‘ ํ‘‡ํด = 1, ํ‘‡ํต = ํ‘‘ํ‘–ํ‘Žํ‘” ํ‘’ , โ€ฆ , ํ‘’ , ฯƒํ‘„๐‘– = 0

It is possible to go from the Lax matrix of a bundle to another by means of a singular gauge transformation (Hecke modification of the bundle)

For the first nontrivial case of one puncture, ํ‘ = 2, the Lax matrix is that of the elliptic Calogero-Moser integrable system:

โ€ฒ ํ‘ ํ‘–ํ‘”ํ‘ฅ(ํ‘„, ํ‘ง) ํœƒ1 ํ‘ง โˆ’ ํ‘„ ํœ ํœƒ1 ํœ ํฟ ํ‘ง ํœ = ํ‘ฅ ํ‘„, ํ‘ง = ํ‘–ํ‘”ํ‘ฅ(โˆ’ํ‘„, ํ‘ง) โˆ’ํ‘ ํœƒ1 ํ‘ง ํœ ํœƒ1(โˆ’ํ‘„)

ํ‘‘2ํ‘„ = 2ํœ‹ํ‘–ํ‘š 2โ„˜โ€ฒ 2ํ‘„ ํœ ํ‘‘ํœ2 ํœ• ํ‘Œ ํ‘ง ํœ = ํฟ ํ‘ง ํœ ํ‘Œ ํ‘ง ํœ 1 ํ‘ง ํฟ2 = ํ‘Œโˆ’1ํœ• ํ‘Œ 2 2 ํ‘ง

GOAL ONE: Find a matrix function Y with given singular behavior and monodromies around points ํ‘งํ‘˜, monodromies and twists around A- and B-cycle. GOAL TWO: Obtain from Y the tau function generating isomonodromic Hamiltonians. (deautonomization of quadratic Hitchin Hamiltonians)

1 1 1 ํป = เถป ํ‘‘ํ‘ง ํ‘กํ‘Ÿ ํฟ2(ํ‘ง) = ํœ• log ํ’ฏ 2 ํ‘˜ 2ํœ‹ํ‘– 2 ํ‘กํ‘˜ ํปํœ = เถป ํ‘‘ํ‘ง ํ‘กํ‘Ÿ ํฟ (ํ‘ง) = 2ํœ‹ํ‘–ํœ•ํœ log ํ’ฏ ํ›พํ‘˜ ํด 2 CFT approach to Mathematical Physical Isomonodromic Conclusions and Formulation of motivation deformations on Outlook the problem the torus Main tool: degenerate braiding and fusion I

For the 1-punctured torus we need only the following braiding move (analytic continuation around a point):

In general [Moore,Seiberg 1989; Alday et al. 2010; Drukker, Teschner et al. 2010]

BRAIDING

FUSION One Punctured Torus: Solution to the Linear System I

ํœ™ โŠ— ํ‘‰ = ํ‘‰ 1 โŠ• ํ‘‰ 1 , ํœ™ , ํ‘  = 1,2 Chiral conformal block [ILT2015]: ํ‘š ํ‘š+ ํ‘šโˆ’ ํ‘  2 2

tr ํ‘žํฟ0ํ‘‰ 1 ํœ™เทจ ํ‘ง โŠ— ํœ™ ํ‘ง ํ‘‰ํ‘š(1)ํœ™เทจ ํ‘ง0 โŠ— ํœ™ ํ‘ง ํ’ฑํ‘Ž ํ‘š 0 ฮฆ ํ‘ง, ํ‘ง0 ํœ, ํ‘Ž, ํ‘š = = โŸจํ‘‰ โŸฉ ํฟ0 ํ‘š trํ’ฑํ‘Ž ํ‘ž ํ‘‰ํ‘š 1

๐‘–ํ‘›ํœ‚ โ€ข ฯƒํ‘› ํ‘’ ฮฆ(ํ‘Ž + ํ‘›) would have given monodromies as in the sphere around 1 and A-cycle:

2ํœ‹๐‘–ํ’‚ 2ํœ‹๐‘–ํ’Ž ํ‘€ํด โˆผ ํ‘’ ํ‘€1 โˆผ ํ‘’

โ€ข But B-cycle has half-integer shifts! โ€ข Necessary to add U(1) boson and consider free fermions

ํœ“ ํ‘ง = ํœ™ ํ‘ง ํ‘’๐‘–ํœ‘ ํ‘ง One Punctured Torus: Solution to the Linear System I

ํœ™ โŠ— ํ‘‰ = ํ‘‰ 1 โŠ• ํ‘‰ 1 , ํœ™ , ํ‘  = 1,2 Chiral conformal block [ILT2015]: ํ‘š ํ‘š+ ํ‘šโˆ’ ํ‘  2 2

tr ํ‘žํฟ0ํ‘‰ 1 ํœ™เทจ ํ‘ง โŠ— ํœ™ ํ‘ง ํ‘‰ํ‘š(1)ํœ™เทจ ํ‘ง0 โŠ— ํœ™ ํ‘ง ํ’ฑํ‘Ž ํ‘š 0 ฮฆ ํ‘ง, ํ‘ง0 ํœ, ํ‘Ž, ํ‘š = = โŸจํ‘‰ โŸฉ ํฟ0 ํ‘š trํ’ฑํ‘Ž ํ‘ž ํ‘‰ํ‘š 1

๐‘–ํ‘›ํœ‚ โ€ข ฯƒํ‘› ํ‘’ ฮฆ(ํ‘Ž + ํ‘›) would have given monodromies as in the sphere around 1 and A-cycle:

2ํœ‹๐‘–ํ’‚ 2ํœ‹๐‘–ํ’Ž ํ‘€ํด โˆผ ํ‘’ ํ‘€1 โˆผ ํ‘’

โ€ข But B-cycle has half-integer shifts! โ€ข Necessary to add U(1) boson and consider free fermions เดค โ€ฒ ํ‘‰ํ‘š 1 ํœ“ ํ‘ง0 โŠ— ํœ“ ํ‘ง โˆ’1 ํœƒ1 ํ‘ง โˆ’ ํ‘ง0 + ํ‘ธ|ํœ ํœƒ1 ํœ Kernel: ํพ ํ‘ง, ํ‘ง0 ํœ, ํ‘Ž, ํ‘š โ‰ก = ํ‘Œ ํ‘ง0 ํ‘Œ(ํ‘ง) ํ‘‰ํ‘š ํœƒ1 ํ‘ง โˆ’ ํ‘ง0 ํœƒ1 ํ‘ธ One-Punctured Torus: Tau Function

2 To get the tau function we expand in ํ‘ง โˆ’ ํ‘ง0 ํœƒ ํ‘„ ํœ ํ‘ ํ‘„ = 1 ํ‘กํ‘ค๐‘–ํ‘ ํ‘ก ํœ‚ ํœ 2 โ€ฒ ํœƒ1 ํ‘ง โˆ’ ํ‘ง0 + ํ‘ธ|ํœ ํœƒ1 ํœ 2 1 ํพ ํ‘ง, ํ‘ง = ํ‘Œโˆ’1 ํ‘ง ํ‘Œ(ํ‘ง) 2 0 0 tr ํพ = + ํ‘ง โˆ’ ํ‘ง0 ํ‘กํ‘Ÿ ํฟ ํ‘ง + 2ํœ‹ํ‘–ํœ•ํœ log ํ‘ํ‘กํ‘ค๐‘–ํ‘ ํ‘ก(ํ‘„) ํœƒ1 ํ‘ง โˆ’ ํ‘ง0 ํœƒ1 ํ‘ธ ํ‘ง โˆ’ ํ‘ง0 2 ํœ•ํ‘งํ‘Œ ํ‘ง = ํฟ ํ‘ง ํ‘Œ(ํ‘ง) Additional contribution due to the twists

ํ‘‰ํ‘šํ‘‡ ํ‘ง 2 Fermion OPE: tr ํพ = = ํ‘‡ + ํ‘š โ„˜ ํ‘ง โˆ’ 1 ํœ + 2ํœ‚1 ํœ + 2ํœ‹ํ‘–ํœ•ํœ logโŸจํ‘‰ํ‘šโŸฉ โŸจํ‘‰ํ‘šโŸฉ

2ํœ‹ํ‘–ํœ•ํœ log ํ‘

1 2 2ํœ‹ํ‘– ํœ•ํœ log ํ’ฏ = ํปํœ = เถป ํ‘กํ‘Ÿ ํฟ ํ‘‘ํ‘ง ํด 2

1 1 ๐‘–ํ‘›ํœ‚ 1 ํฟ0 ํท Tau function: ํ’ฏ = ํ‘‰ํ‘š 1 ํท = เท ํ‘’ 2 tr ํ‘› ํ‘ž ํ‘‰ํ‘š 1 โˆ ํ‘ Ztwist ํ‘ํ‘กํ‘ค๐‘–ํ‘ ํ‘ก โ„‹ 2 ํ‘ํ‘กํ‘ค๐‘–ํ‘ ํ‘ก ํ‘› ํ‘Ž Generalizations

The generalization from one to many punctures is straightforward: one only has to add a primary field insertion at every punctureโ€™s location.

We dealt with the case of an ํ‘†ํฟ(2) linear system. The picture can be extended to the case of ํ‘†ํฟ(ํ‘) Fuchsian systems by using a ํ‘Šํ‘ of free fermions. [BDGT19xx.xxxx]

โ€ฒ เดค โˆ’1 ํœƒ1 ํ‘ง โˆ’ ํ‘ง0 + ํ‘ธ|ํœ ํœƒ1 ํœ ํ‘‰1 ํ‘ง1 โ€ฆ ํ‘‰ํ‘› ํ‘งํ‘› ํœ“ ํ‘ง0 โŠ— ํœ“ ํ‘ง ํ‘†ํฟ(ํ‘) Kernel ํ‘Œ ํ‘ง0 Y z = ํœƒ1 ํ‘ง โˆ’ ํ‘ง0 ํœƒ1 ํ‘ธ โŸจํ‘‰1 โ€ฆ ํ‘‰ํ‘›โŸฉ

1 ํ’ฏ = โŸจํ‘‰ ํ‘ง โ€ฆ ํ‘‰ ํ‘ง โŸฉ ํ‘†ํฟ(ํ‘) Tau function ํœƒ ํ‘„ ํœ 1 1 ํ‘› ํ‘› ฯ‚ 1 ๐‘– ๐‘– ํœ‚(ํœ) Contents

CFT approach to Mathematical Physical Isomonodromic Conclusions and Formulation of motivation deformations on Outlook the problem the torus Conclusions and Outlook

โ€ข We extended the link between isomonodromy deformations, two-dimensional CFT and four- dimensional ํ’ฉ = 2 gauge theories to the case of genus 1. New ingredients are needed because of nontriviality of bundles on the torus. [1901.10497]

โ€ข The construction can be extended to higher number of punctures and ํ‘†ํฟํ‘ linear systems [BDGT, to appear]

โ€ข Higher genus extensions: what is the physical meaning of ํ‘ํ‘กํ‘ค๐‘–ํ‘ ํ‘ก? What is its analogue for ํ‘” > 1?

โ€ข Connection with surface operators, B-branes and KZ equations/quantum isomonodromy (work in progress)

โ€ข Uplift to 5d: q-deformation of isomonodromy problem on the torus, connection with topological strings and quantum mirror curves [GHM2016; BGT 2017,2018; CPT 2018]

โ€ข Mathematical proof of Fredholm determinant expression following the approach of [Gavrylenko Lisovyy, Cafasso Gavrylenko Lisovyy 2018] Thank you!