Isomonodromic $\tau$-functions and $W_N$ conformal blocks

Abstract

We study the solution of the Schlesinger system for the 4-point ${\mathfrak{sl}}_N$ isomonodromy problem and conjecture an expression for the isomonodromic $\tau$-function in terms of 2d conformal field theory beyond the known $N=2$ Painlevé VI case. We show that this relation can be used as an alternative definition of conformal blocks for the $W_N$ algebra and argue that the infinite number of arbitrary constants arising in the algebraic construction of $W_N$ conformal block can be expressed in terms of only a finite set of parameters of the monodromy data of rank $N$ Fuchsian system with three regular singular points. We check this definition explicitly for the known conformal blocks of the $W_3$ algebra and demonstrate its consistency with the conjectured form of the structure constants.

22 pages, 7 figures; version to appear in JHEP

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Cited by in corpus (21)

• On Painlevé/gauge theory correspondence
• Fredholm determinant and Nekrasov sum representations of isomonodromic tau functions
• Monodromy dependence and connection formulae for isomonodromic tau functions
• Cluster integrable systems, q-Painleve equations and their quantization
• Kerr-de Sitter Greybody Factors via Isomonodromy
• A slow review of the AGT correspondence
• Riemann-Hilbert correspondence and blown up surface defects
• General Results for Higher Spin Wilson Lines and Entanglement in Vasiliev Theory
• 2-parameter $\tau$-function for the first Painlevé equation -Topological recursion and direct monodromy problem via exact WKB analysis-
• Cluster Toda chains and Nekrasov functions
• $\mathcal{N}=2^{\ast }$ gauge theory, free fermions on the torus and Painlevé VI
• Free fermions, W-algebras and isomonodromic deformations
• Quantum spectral problems and isomonodromic deformations
• Higher rank isomonodromic deformations and W-algebras
• Exact conformal blocks for the W-algebras, twist fields and isomonodromic deformations
• Rigid Fuchsian systems in 2-dimensional conformal field theories
• Twist-field representations of W-algebras, exact conformal blocks and character identities
• Crossing invariant correlation functions at $c=1$ from isomonodromic $\tau$ functions
• Circular quiver gauge theories, isomonodromic deformations and $W_N$ fermions on the torus
• The geometry of Casimir W-algebras
• Topological string amplitudes and Seiberg-Witten prepotentials from the counting of dimers in transverse flux