Painlevé Kernels and Surface Defects at Strong Coupling

Abstract

It is well established that the spectral analysis of canonically quantized four-dimensional Seiberg-Witten curves can be systematically studied via the Nekrasov-Shatashvili functions. In this paper, we explore another aspect of the relation between $\mathcal{N}=2$ supersymmetric gauge theories in four dimensions and operator theory. Specifically, we study an example of an integral operator associated with Painlevé equations and whose spectral traces are related to correlation functions of the 2d Ising model. This operator does not correspond to a canonically quantized Seiberg-Witten curve, but its kernel can nevertheless be interpreted as the density matrix of an ideal Fermi gas. Adopting the approach of Tracy and Widom, we provide an explicit expression for its eigenfunctions via an $\mathrm{O}(2)$ matrix model. We then show that these eigenfunctions are computed by surface defects in $\mathrm{SU}(2)$ super Yang-Mills in the self-dual phase of the $\Omega$-background. Our result also yields a strong coupling expression for such defects which resums the instanton expansion. Even though we focus on one concrete example, we expect these results to hold for a larger class of operators arising in the context of isomonodromic deformation equations.

arXiv commentVersion 2: 50 pages, 1 figure. Minor corrections and clarifications, additional references. Comments welcome

References in corpus (110)

• Monopole Condensation, And Confinement In N=2 Supersymmetric Yang-Mills Theory
• Monopoles, Duality and Chiral Symmetry Breaking in N=2 Supersymmetric QCD
• Liouville Correlation Functions from Four-dimensional Gauge Theories
• Geometric Engineering of Quantum Field Theories
• Topological Strings and Integrable Hierarchies
• The Refined Topological Vertex
• Holomorphic Anomalies in Topological Field Theories
• Loop and surface operators in N=2 gauge theory and Liouville modular geometry
• Instantons and recursion relations in N=2 Susy gauge theory
• Self-Dual Strings and N=2 Supersymmetric Field Theory
• Discrete Heisenberg-Weyl Group and Modular Group
• Integrating Over Higgs Branches
• Vortex Counting and Lagrangian 3-manifolds
• Nekrasov Functions and Exact Bohr-Sommerfeld Integrals
• Quantization of Integrable Systems and Four Dimensional Gauge Theories
• A New Supersymmetric Index
• Disk Instantons, Mirror Symmetry and the Duality Web
• Open string amplitudes and large order behavior in topological string theory
• Multi-Instanton Calculus and Equivariant Cohomology
• Non-perturbative effects and the refined topological string
• An Algorithm for the Microscopic Evaluation of the Coefficients of the Seiberg-Witten Prepotential
• Issues in Topological Gauge Theory
• Darboux coordinates, Yang-Yang functional, and gauge theory
• Remodeling the B-model
• Gauge Theory Loop Operators and Liouville Theory
• Quantum Geometry of Refined Topological Strings
• Instanton Effects in ABJM Theory from Fermi Gas Approach
• Defects and Quantum Seiberg-Witten Geometry
• Affine SL(2) conformal blocks from 4d gauge theories
• Refined Topological Vertex and Instanton Counting
• A & B model approaches to surface operators and Toda theories
• Painleve III and 2D Polymers
• Exact Solution of the O(n) Model on a Random Lattice
• Nekrasov Functions from Exact BS Periods: the Case of SU(N)
• Exact Results on the ABJM Fermi Gas
• Matone's Relation in the Presence of Gravitational Couplings
• Irregular Liouville correlators and connection formulae for Heun functions
• Multicritical Phases of the O(n) Model on a Random Lattice
• Instanton counting with a surface operator and the chain-saw quiver
• Exact vs. Semiclassical Target Space of the Minimal String
• How instanton combinatorics solves Painlevé VI, V and III's
• Holomorphic Anomaly in Gauge Theories and Matrix Models
• Isomonodromic tau-functions from Liouville conformal blocks
• Surface Defects and Resolvents
• A-polynomial, B-model, and Quantization
• Matrix models from operators and topological strings, 2
• On Painlevé/gauge theory correspondence
• N=2 supersymmetric dynamics for pedestrians
• More on the exact solution of the O(n) model on a random lattice and an investigation of the case |n|>2
• New Exact Quantization Condition for Toric Calabi-Yau Geometries
• Operators from mirror curves and the quantum dilogarithm
• Exact Solution of the Six-Vertex Model on a Random Lattice
• Fredholm determinant and Nekrasov sum representations of isomonodromic tau functions
• Surface Operator, Bubbling Calabi-Yau and AGT Relation
• A holomorphic and background independent partition function for matrix models and topological strings
• On the spectrum of Dehn twists in quantum Teichmuller theory
• Non-perturbative approaches to the quantum Seiberg-Witten curve
• M-theoretic matrix models
• Exact quantization conditions for the relativistic Toda lattice
• Opers, surface defects, and Yang-Yang functional
• Proofs of Two Conjectures Related to the Thermodynamic Bethe Ansatz
• Nekrasov prepotential with fundamental matter from the quantum spin chain
• Bilinear equations on Painleve tau functions from CFT
• Integrability and cycles of deformed $\mathcal{N}=2$ gauge theory
• Connection problem for the sine-Gordon/Painlevé III tau function and irregular conformal blocks
• Q-deformed Painleve tau function and q-deformed conformal blocks
• Geometry of Spectral Curves and All Order Dispersive Integrable System
• Operators and higher genus mirror curves
• Perturbative connection formulas for Heun equations
• ABJM on ellipsoid and topological strings
• The Wave Function Behavior of the Open Topological String Partition Function on the Conifold
• Irregular conformal blocks, with an application to the fifth and fourth Painlevé equations
• Exact WKB and abelianization for the $T_3$ equation
• TBA for the Toda chain
• Intersecting Surface Defects and Instanton Partition Functions
• Surface defects and instanton-vortex interaction
• A Solvable Deformation of Quantum Mechanics
• Quantization condition from exact WKB for difference equations
• Riemann-Hilbert correspondence and blown up surface defects
• Exact Quantization Conditions, Toric Calabi-Yau and Nonperturbative Topological String
• Asymptotics of a Class of Solutions to the Cylindrical Toda Equations
• Splitting of surface defect partition functions and integrable systems
• 2-parameter $\tau$-function for the first Painlevé equation -Topological recursion and direct monodromy problem via exact WKB analysis-
• Exact eigenfunctions and the open topological string
• Quantum Curve and the First Painlevé Equation
• Surface operators, chiral rings, and localization in N=2 gauge theories
• Classical Conformal Blocks and Accessory Parameters from Isomonodromic Deformations
• Cluster Toda chains and Nekrasov functions
• Enumeration of maps with self avoiding loops and the O(n) model on random lattices of all topologies
• Exact relativistic Toda chain eigenfunctions from Separation of Variables and gauge theory
• BPS relations from spectral problems and blowup equations
• Some Classes of Solutions to the Toda Lattice Hierarchy
• Exact WKB and the quantum Seiberg-Witten curve for 4d $N=2$ pure $SU(3)$ Yang-Mills, Part I: Abelianization
• Pure $SU(2)$ gauge theory partition function and generalized Bessel kernel
• On the connection problem for Painlevé I
• Isomonodromic deformations of a rational differential system and reconstruction with the topological recursion: the ${\mathfrak{sl}}_2$ case
• Modular and duality properties of surface operators in N=2* gauge theories
• Quantum periods and prepotential in $\mathcal{N}=2$ SU(2) SQCD
• On Large N Solution of N=3 Chern-Simons-adjoint Theories
• Quantum spectral problems and isomonodromic deformations
• Wall-crossing of TBA equations and WKB periods for the third order ODE
• Isomonodromic tau functions on a torus as Fredholm determinants, and charged partitions
• Irregular conformal blocks, Painlevé III and the blow-up equations
• Quantum periods and TBA equations for $\mathcal{N}=2SU(2)N_f=2$ SQCD with flavor symmetry
• Relations between Stokes constants of unrefined and Nekrasov-Shatashvili topological strings
• $t{t}^{\ast }$ Toda equations for surface defects in $\mathcal{N}=2$ SYM and instanton counting for classical Lie groups
• Remarks on irregular conformal blocks and Painlevé III and II tau functions
• Parallel surface defects, Hecke operators, and quantum Hitchin system
• Fredholm determinant representation of the Painlevé II $\tau$-function
• Connecting topological strings and spectral theory via non-autonomous Toda equations