arXiv subjects

A. I. Shchechkin

Publications and source records attributed to A. I. Shchechkin.

3 recordsLinked to original sources

arXiv2019

Q-deformed Painleve tau function and q-deformed conformal blocks

We propose $q$-deformation of the Gamayun-Iorgov-Lisovyy formula for Painlevé $τ$ function. Namely we propose formula for $τ$ function for $q$-difference Painlevé equation corresponding to $A_7^{(1)}{}'$ surface (and $A_1^{(1)}$ symmetry) in Sakai's classification. In this formula $τ$ function equals the series of $q$-Virasoro Whittaker conformal blocks (equivalently Nekrasov partition functions for pure $SU(2)$ 5d theory).

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arXiv2017

Backlund transformation of Painleve III($D_8$) tau function

We study explicit formula (suggested by Gamayun, Iorgov, Lisovyy) for Painlevé III($D_8$) $τ$ function in terms of Virasoro conformal blocks with central charge $1$. The Painlevé equation has two types of bilinear forms, we call them Toda-like and Okamoto-like. We obtain these equations from the representation theory using an embedding of a direct sum of two Virasoro algebra in a certain superalgebra. These two types of bilinear forms correspond to Neveu-Schwarz sector and Ramond sector of this algebra. We also obtain $τ$ functions of algebraic solutions of Painlevé III($D_8$) from the special representations of the Virasoro algebra of highest weight $(n+1/4)^2$.

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arXiv2015

Bilinear equations on Painleve tau functions from CFT

In 2012 Gamayun, Iorgov, Lisovyy conjectured an explicit expression for the Painlevé VI $τ$~function in terms of the Liouville conformal blocks with central charge $c=1$. We prove that proposed expression satisfies Painlevé VI $τ$~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).

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