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Tau Function 풯 = 푍 = Σ푛 푍 푎 + 푛 푒 of the Corresponding Gauge Theory in the Self-Dual Omega Background Topological Strings and Hitchin Systems

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Tau Function 풯 = 푍 = Σ푛 푍 푎 + 푛 푒 of the Corresponding Gauge Theory in the Self-Dual Omega Background Topological Strings and Hitchin Systems Painlevé/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 5푑 풩 = 2 퐶,푛,휃 target Hitchin 푆푌푀(퐺) 푆1 moduli space 풩 = (0,2) Class 1 3d NLSM with 풮(픤) 푆 퐶 풮(픤, 퐶 ) target SW ,푛,휃 ,푛,휃 moduli space Class S theories and Hitchin system • Data for the 4d theory: 퐶,푛,휃: 푔, 푛 determines the gauge and matter content; 휃푘, 휎 determine masses of hypermultiplets and vector vev 휗Ԧ3 휗Ԧ1 휗Ԧ2 휗Ԧ4 Weak coupling regime Degeneration limit of RS • Hitchin system: Lax pair 퐿, 푀 meromorphic matrix-valued differentials on 퐶,푛 푘 • 푡푟 퐿 = 푢푘 Hamiltonians of integrable system/Coulomb branch parameters; double pole 2 coefficients of 푡푟 퐿 are 휃푘’s • Σ: det(퐿 − 휆퐼) is the SW curve Isomonodromic deformations and class S theories Complex structure moduli of 퐶,푛 Gauge couplings (or dynamical energy scale in the asymptotically free cases) in weakly coupled frames • 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) • These deformations are well-known from the point of view of integrable systems: they are isomonodromic deformations of the Hitchin system • These flows are Hamiltonian, the generating function for these Hamiltonians is 퐷 푛휂 called the tau function 풯 = 푍 = σ푛 푍 푎 + 푛 푒 of the corresponding gauge theory in the self-dual Omega background Topological Strings and Hitchin Systems TS • Open CY3 X Mirror curve Σ푋 (nontrivial part of the mirror CY3) 푇푆 • 4d limit of Σ푋 is the spectral curve ΣX: det 퐿 − 휆퐼 = 0. • The isomonodromic deformations of the Hitchin system turn out to be closely related to the quantization of the spectral curve. • The differential equations are exactly satisfied by the dual gauge theory partition function in self-dual Omega background (unrefined TS) [GIL2013,ILT2015,GM2016, BLMST2017] • If one considers the problem before the 4d limit, the “deformation” equations are really difference equations (q-Painlevé) satisfied by the TS partition function [BGT2018,BGM2018] • The tau function can be written as a Fredholm determinant [Z1994,GL2018,CGL2018]: nonperturbative definition of TS! Isomonodromic deformations and quantum mirror curves • BGT → GHM: inverse of quantum mirror curve is a trace class operator (see A. Grassi’s talk) • One can show its Fredholm determinant in the 4d limit is tau function of an isomonodromic problem (pure gauge, 4d and 5d) • 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. • CPT (see J. Teschner’s talk): • The free fermion representation of the partition function provides the bridge between TS and integrable hierarchies, again the Fredholm determinant provides nonperturbative definition • Follows the approach of [ADKMV2005,DHSV2008]: free fermion conformal blocks correspond to B-model amplitudes with possible insertion of B-branes. • 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 • Extend this correspondence to the (differential, 4d) case of g=1 with n regular punctures • 4d gauge theory: circular quiver gauge theories, with matter in the adjoint representation of the gauge group (6d theory little strings, see S. Hohenegger’s talk) • Mathematical physics: explicit solution of isomonodromic problems on higher genus Riemann surfaces is still an open problem Results I • On the torus, there is an extra factor between tau function and NO dual partition function/free fermion chiral correlator:< 퐷 풯푠푝ℎ푒푟푒 = 푍 • One puncture case: 퐷 퐷 퐷 푍 푍 푍 Τ 풯 = = 0 = 1 2 2 휃 2푄 휃 2푄 휃1 푄 3 2 휂 휏 2 휂 휏 휂 휏 푑2푄 = 2휋푖푚 2℘′ 2푄 휏 (Manin’s PVI special case) 푑휏2 Results I • On the torus, there is an extra factor between tau function and NO dual partition function/free fermion chiral correlator: 퐷 푍 푁 휃1 푄푖 풯 = 푍푡푤푠푡 = ς=1 푍푡푤푖푠푡 푄 휂 휏 • One puncture case: 퐷 퐷 퐷 푍 푍 푍 Τ 풯 = = 0 = 1 2 2 휃 2푄 휃 2푄 휃1 푄 3 2 휂 휏 2 휂 휏 휂 휏 푑2푄 = 2휋푖푚 2℘′ 2푄 휏 (Manin’s PVI special case) 푑휏2 Results II • The last equation gives us an exact relation between the Painlevé 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 퐷 푍0 휃3 2푄 휏 2휏 퐷 = 푍1Τ2 휃2(2푄(휏)|2휏) 2 2 2 2 2 2 8 휃2 2휏푆푊 휃3 2휏푆푊 휃2 2휏 휃3 2휏 휋 휇 휃4 2휏 2 + 2 = 2 + 2 − 2 2 2 휃3 2휏푆푊 휃2 2휏푆푊 휃3 2휏 휃2 2휏 푢 − 4휋푖휇 휕휏 log 휃2 휏 휃2 2휏 휃3 2휏 • Fredholm determinant 2 representation of the (dual) 푍퐷 = 푁 푎, 푚, 푎 푞푎 det(1 + 퐾) 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 −휗푘 푌 푧~푧푘 = 퐺푘 푧 − 푧푘 퐶푘 휕푧푌 푧 = 퐿 푧 푌 푧 푛 ൞ det 푌 푧0 ≠ 0 퐴푘 푌 푧0 = 푖푑푁 퐿 푧 = ෎ 푧 − 푧푘 • 휗Ԧ푘 are the local monodromy exponents 푘=1 푌 훾푘 ⋅ 푧 = 푌 푧 푀푘 • Kernel: 푌 푧0 = 푖푑푁 푌−1 푧 푌 푧 0 퐴 = 퐶 휽 퐶−1 퐾 푧, 푧0 = 푘 푘 풌 푘 푧 − 푧0 2휋휽풌 −1 푀퐾 = 퐶푘푒 퐶푘 Isomonodromic deformations: Deformations of the above linear systems that preserve the conjugacy classes of the monodromies (moving the points 푧푘) 1 1 2 Isomonodromic tau function: 휕푧 log 풯 = 퐻푘, 퐻푘 = ׯ 푡푟퐿 푑푧 푘 2휋 훾푘 2 Linear Systems on the Torus On the torus there cannot be a • Take 퐿(푧) to have 푁 additional singularities with residues of rank 1 function with just one simple pole, [Krichever 2002]; so one cannot directly generalize 퐿 푧 . • Consider 퐿(푧) to be a section of a nontrivial bundle on the torus There is more than one possibility: [Korotkin, Takasaki 1997-2003]. 2휋푖푘 Flat 푆퐿(푁, ℂ) bundles on the torus can be classified by an element 푒 푁 of their center ℤ푁, i.e. by an integer 푘 = 0, … , 푁 − 1 through their transition functions [Levin et al. 2014] 2휋푘 −1 −1 −1 −1 푁 퐿 푧 + 1 = 푇퐴퐿 푧 푇퐴 , 퐿 푧 + 휏 = 푇퐵퐿 푧 푇퐵 , 푇퐴푇퐵푇퐴 푇퐵 = 푒 Both twists and monodromies 휕푧푌 = 퐿푌 푌 → 푇푌푀 Linear System on the Torus Bundles with 푘 = 0 2휋푄1 2휋푄푁 푇퐴 = 1, 푇퐵 = 푑푖푎푔 푒 , … , 푒 , σ푄 = 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, 푁 = 2, the Lax matrix is that of the elliptic Calogero-Moser integrable system: ′ 푝 푖푔푥(푄, 푧) 휃1 푧 − 푄 휏 휃1 휏 퐿 푧 휏 = 푥 푄, 푧 = 푖푔푥(−푄, 푧) −푝 휃1 푧 휏 휃1(−푄) 푑2푄 = 2휋푖푚 2℘′ 2푄 휏 푑휏2 휕 푌 푧 휏 = 퐿 푧 휏 푌 푧 휏 1 푧 퐿2 = 푌−1휕 푌 2 2 푧 GOAL ONE: Find a matrix function Y with given singular behavior and monodromies around points 푧푘, 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 퐻 = ර 푑푧 푡푟 퐿2(푧) = 휕 log 풯 2 푘 2휋푖 2 푡푘 퐻휏 = ර 푑푧 푡푟 퐿 (푧) = 2휋푖휕휏 log 풯 훾푘 퐴 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 휙 ⊗ 푉 = 푉 1 ⊕ 푉 1 , 휙 , 푠 = 1,2 Chiral conformal block [ILT2015]: 푚 푚+ 푚− 푠 2 2 tr 푞퐿0푉 1 휙෨ 푧 ⊗ 휙 푧 푉푚(1)휙෨ 푧0 ⊗ 휙 푧 풱푎 푚 0 Φ 푧, 푧0 휏, 푎, 푚 = = ⟨푉 ⟩ 퐿0 푚 tr풱푎 푞 푉푚 1 푛휂 • σ푛 푒 Φ(푎 + 푛) would have given monodromies as in the sphere around 1 and A-cycle: 2휋풂 2휋풎 푀퐴 ∼ 푒 푀1 ∼ 푒 • But B-cycle has half-integer shifts! • Necessary to add U(1) boson and consider free fermions 휓 푧 = 휙 푧 푒휑 푧 One Punctured Torus: Solution to the Linear System I 휙 ⊗ 푉 = 푉 1 ⊕ 푉 1 , 휙 , 푠 = 1,2 Chiral conformal block [ILT2015]: 푚 푚+ 푚− 푠 2 2 tr 푞퐿0푉 1 휙෨ 푧 ⊗ 휙 푧 푉푚(1)휙෨ 푧0 ⊗ 휙 푧 풱푎 푚 0 Φ 푧, 푧0 휏, 푎, 푚 = = ⟨푉 ⟩ 퐿0 푚 tr풱푎 푞 푉푚 1 푛휂 • σ푛 푒 Φ(푎 + 푛) would have given monodromies as in the sphere around 1 and A-cycle: 2휋풂 2휋풎 푀퐴 ∼ 푒 푀1 ∼ 푒 • But B-cycle has half-integer shifts! • Necessary to add U(1) boson and consider free fermions ത ′ 푉푚 1 휓 푧0 ⊗ 휓 푧 −1 휃1 푧 − 푧0 + 푸|휏 휃1 휏 Kernel: 퐾 푧, 푧0 휏, 푎, 푚 ≡ = 푌 푧0 푌(푧) 푉푚 휃1 푧 − 푧0 휃1 푸 One-Punctured Torus: Tau Function 2 To get the tau function we expand in 푧 − 푧0 휃 푄 휏 푍 푄 = 1 푡푤푠푡 휂 휏 2 ′ 휃1 푧 − 푧0 + 푸|휏 휃1 휏 2 1 퐾 푧, 푧 = 푌−1 푧 푌(푧) 2 0 0 tr 퐾 = + 푧 − 푧0 푡푟 퐿 푧 + 2휋푖휕휏 log 푍푡푤푠푡(푄) 휃1 푧 − 푧0 휃1 푸 푧 − 푧0 2 휕푧푌 푧 = 퐿 푧 푌(푧) Additional contribution due to the twists 푉푚푇 푧 2 Fermion OPE: tr 퐾 = = 푇 + 푚 ℘ 푧 − 1 휏 + 2휂1 휏 + 2휋푖휕휏 log⟨푉푚⟩ ⟨푉푚⟩ 2휋푖휕휏 log 푍 1 2 2휋푖 휕휏 log 풯 = 퐻휏 = ර 푡푟 퐿 푑푧 퐴 2 1 1 푛휂 1 퐿0 퐷 Tau function: 풯 = 푉푚 1 퐷 = ෍ 푒 2 tr 푛 푞 푉푚 1 ∝ 푍 Ztwist 푍푡푤푠푡 ℋ 2 푍푡푤푠푡 푛 푎 Generalizations The generalization from one to many punctures is straightforward: one only has to add a primary field insertion at every puncture’s location.
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