Fredholm determinant and Nekrasov sum representations of isomonodromic tau functions

Abstract

We derive Fredholm determinant representation for isomonodromic tau functions of Fuchsian systems with $n$ regular singular points on the Riemann sphere and generic monodromy in $\mathrm{GL}(N,\mathbb{C})$. The corresponding operator acts in the direct sum of $N(n-3)$ copies of $L^2(S^1)$. Its kernel has a block integrable form and is expressed in terms of fundamental solutions of $n-2$ elementary 3-point Fuchsian systems whose monodromy is determined by monodromy of the relevant $n$-point system via a decomposition of the punctured sphere into pairs of pants. For $N=2$ these building blocks have hypergeometric representations, the kernel becomes completely explicit and has Cauchy type. In this case Fredholm determinant expansion yields multivariate series representation for the tau function of the Garnier system, obtained earlier via its identification with Fourier transform of Liouville conformal block (or a dual Nekrasov-Okounkov partition function). Further specialization to $n=4$ gives a series representation of the general solution to Painlevé VI equation.

arXiv comment45 pages, 16 figures; v3: minor changes and added refs to match the published version

References in corpus (31)

• Level-Spacing Distributions and the Airy Kernel
• Liouville Correlation Functions from Four-dimensional Gauge Theories
• Fredholm Determinants, Differential Equations and Matrix Models
• Level-Spacing Distributions and the Airy Kernel
• On combinatorial expansion of the conformal blocks arising from AGT conjecture
• A Quantum Deformation of the Virasoro Algebra and the Macdonald Symmetric Functions
• Asymptotically free N=2 theories and irregular conformal blocks
• Wild Quiver Gauge Theories
• Irregular singularities in Liouville theory
• Non-Lagrangian Theories from Brane Junctions
• How instanton combinatorics solves Painlevé VI, V and III's
• Isomonodromic tau-functions from Liouville conformal blocks
• Conformal field theory of Painlevé VI
• Integrable structure, W-symmetry and AGT relation
• Integrable Fredholm Operators and Dual Isomonodromic Deformations
• Bilinear equations on Painleve tau functions from CFT
• Painlevé VI connection problem and monodromy of c=1 conformal blocks
• Monodromy dependence and connection formulae for isomonodromic tau functions
• Connection problem for the sine-Gordon/Painlevé III tau function and irregular conformal blocks
• Z-Measures on partitions, Robinson-Schensted-Knuth correspondence, and beta=2 random matrix ensembles
• Towards a 4d/2d correspondence for Sicilian quivers
• Irregular conformal blocks, with an application to the fifth and fourth Painlevé equations
• Isomonodromic $\tau$-functions and $W_N$ conformal blocks
• Painlevé monodromy manifolds, decorated character varieties and cluster algebras
• Free fermions, W-algebras and isomonodromic deformations
• Deformation analysis of matrix models
• Fredholm determinants, Jimbo-Miwa-Ueno tau-functions, and representation theory
• Exact conformal blocks for the W-algebras, twist fields and isomonodromic deformations
• Analytic solutions of $q$-$P(A_1)$ near its critical points
• Discrete matrix models for partial sums of conformal blocks associated to Painlevé transcendents
• Isomonodromic deformations in genus zero and one: algebrogeometric solutions and Schlesinger transformations

Cited by in corpus (47)

• On Painlevé/gauge theory correspondence
• Monodromy dependence and connection formulae for isomonodromic tau functions
• Irregular conformal blocks and connection formulae for Painlevé V functions
• Kerr-de Sitter Quasinormal Modes via Accessory Parameter Expansion
• A slow review of the AGT correspondence
• Scalar quasinormal modes of Kerr-AdS${}_5$
• On the analysis of incomplete spectra in random matrix theory through an extension of the Jimbo-Miwa-Ueno differential
• Confluent conformal blocks and the Teukolsky master equation
• 2-parameter $\tau$-function for the first Painlevé equation -Topological recursion and direct monodromy problem via exact WKB analysis-
• On determinant representation and integrability of Nekrasov functions
• Teukolsky master equation and Painlevé transcendents: numerics and extremal limit
• Classical Conformal Blocks and Accessory Parameters from Isomonodromic Deformations
• Painleve equations from Nakajima-Yoshioka blowup relations
• On some Hamiltonian properties of the isomonodromic tau functions
• $\mathcal{N}=2^{\ast }$ gauge theory, free fermions on the torus and Painlevé VI
• Scalar quasi-normal modes of accelerating Kerr-Newman-AdS black holes
• Argyres-Douglas theories and Liouville Irregular States
• CFT approach to the $q$-Painlevé VI equation
• Quantum spectral problems and isomonodromic deformations
• Orthogonal Polynomials in Mathematical Physics
• Resurgence, Painleve Equations and Conformal Blocks
• q-Painleve equation from Virasoro constraints
• Isomonodromic tau functions on a torus as Fredholm determinants, and charged partitions
• Higher rank isomonodromic deformations and W-algebras
• Hamiltonian structure of rational isomonodromic deformation systems
• Expansions for semiclassical conformal blocks
• On solutions of the Fuji-Suzuki-Tsuda system
• New results in $\mathcal{N}=2$ theories from non-perturbative string
• Monodromy dependence and symplectic geometry of isomonodromic tau functions on the torus
• Twisted Representations of Algebra of $q$-Difference Operators, Twisted $q$-$W$ Algebras and Conformal Blocks
• Fredholm determinant representation of the Painlevé II $\tau$-function
• The $\tau$-function of the Ablowitz-Segur family of solutions to Painlevé II as a Widom constant
• Remarks on holographic models of the Kerr-AdS${}_5$ geometry
• Fifty Years of KdV: An Integrable System
• The Malgrange Form and Fredholm Determinants
• Vector perturbations of Kerr-AdS${}_5$ and the Painlevé VI transcendent
• On the tau function of the hypergeometric equation
• QNMs of scalar fields on small Reissner-Nordström-AdS${}_5$ black holes
• Connection formulas for the lambda generalized Ising correlation functions
• Probing Quantum Curves and Transitions in 5d SQFTs via Defects and Blowup Equations
• On a 5D UV completion of Argyres-Douglas theories
• Painlevé/CFT correspondence on a torus
• Painlevé Kernels and Surface Defects at Strong Coupling
• Topological string amplitudes and Seiberg-Witten prepotentials from the counting of dimers in transverse flux
• Fredholm Pfaffian $\tau$-functions for orthogonal isospectral and isomonodromic systems
• Highest-weight vectors and three-point functions in GKO coset decomposition
• AGT correspondence, (q-)Painlevè equations and matrix models