Bilinear equations on Painleve tau functions from CFT

Abstract

In 2012 Gamayun, Iorgov, Lisovyy conjectured an explicit expression for the Painlevé VI $\tau$~function in terms of the Liouville conformal blocks with central charge $c=1$. We prove that proposed expression satisfies Painlevé VI $\tau$~function bilinear equations (and therefore prove the conjecture). The proof reduces to the proof of bilinear relations on conformal blocks. These relations were studied using the embedding of a direct sum of two Virasoro algebras into a sum of Majorana fermion and Super Virasoro algebra. In the framework of the AGT correspondence the bilinear equations on the conformal blocks can be interpreted in terms of instanton counting on the minimal resolution of ${\mathbb{C}}^2/{\mathbb{Z}}_2$ (similarly to Nakajima-Yoshioka blow-up equations).

35 pages, v2 misprints corrected, v4 references added, misprints corrected, v5 small improvements

References in corpus (3)

• Isomonodromic tau-functions from Liouville conformal blocks
• N=1 SUSY Conformal Block Recursive Relations
• On the N=1 super Liouville four-point functions

Cited by in corpus (34)

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• Cluster integrable systems, q-Painleve equations and their quantization
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• Connection problem for the sine-Gordon/Painlevé III tau function and irregular conformal blocks
• Q-deformed Painleve tau function and q-deformed conformal blocks
• A slow review of the AGT correspondence
• Irregular conformal blocks, with an application to the fifth and fourth Painlevé equations
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• Classical Conformal Blocks and Accessory Parameters from Isomonodromic Deformations
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• Painleve equations from Nakajima-Yoshioka blowup relations
• $\mathcal{N}=2^{\ast }$ gauge theory, free fermions on the torus and Painlevé VI
• Backlund transformation of Painleve III($D_8$) tau function
• Coupling of two conformal field theories and Nakajima-Yoshioka blow-up equations
• Argyres-Douglas theories and Liouville Irregular States
• Quantum spectral problems and isomonodromic deformations
• Spectral determinants and quantum theta functions
• Isomonodromic tau functions on a torus as Fredholm determinants, and charged partitions
• q-Painleve equation from Virasoro constraints
• Higher rank isomonodromic deformations and W-algebras
• Irregular conformal blocks, Painlevé III and the blow-up equations
• Analytic solutions of $q$-$P(A_1)$ near its critical points
• M2-branes and $\mathfrak{q}$-Painlevé equations
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• Discrete matrix models for partial sums of conformal blocks associated to Painlevé transcendents
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• AGT correspondence, (q-)Painlevè equations and matrix models