arXiv subjects

M. A. Bershtein

Publications and source records attributed to M. A. Bershtein.

6 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).

math-ph↗

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 $(n+1/4)^2$.

math-ph↗

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).

math-ph↗

arXiv2013

Bases in coset conformal field theory from AGT correspondence and Macdonald polynomials at the roots of unity

We continue our study of the AGT correspondence between instanton counting on C^2/Z_p and Conformal field theories with the symmetry algebra A(r,p). In the cases r=1, p=2 and r=2, p=2 this algebra specialized to: A(1,2)=H+sl(2)_1 and A(2,2)=H+sl(2)_2+NSR. As the main tool we use a new construction of the algebra A(r,2) as the limit of the toroidal gl(1) algebra for q,t tend to -1. We claim that the basis of the representation of the algebra A(r,2) (or equivalently, of the space of the local fields of the corresponding CFT) can be expressed through Macdonald polynomials with the parameters q,t go to -1. The vertex operator which naturally arises in this construction has factorized matrix elements in this basis. We also argue that the singular vectors of the $\mathcal{N}=1$ Super Virasoro algebra can be realized in terms of Macdonald polynomials for a rectangular Young diagram and parameters q,t tend to -1.

hep-th↗

arXiv2012

Instanton moduli spaces and bases in coset conformal field theory

Recently proposed relation between conformal field theories in two dimensions and supersymmetric gauge theories in four dimensions predicts the existence of the distinguished basis in the space of local fields in CFT. This basis has a number of remarkable properties, one of them is the complete factorization of the coefficients of the operator product expansion. We consider a particular case of the U(r) gauge theory on C^2/Z_p which corresponds to a certain coset conformal field theory and describe the properties of this basis. We argue that in the case p=2, r=2 there exist different bases. We give an explicit construction of one of them. For another basis we propose the formula for matrix elements.

hep-th↗

arXiv2011

Parafermionic polynomials, Selberg integrals and three-point correlation function in parafermionic Liouville field theory

In this paper we consider parafermionic Liouville field theory. We study integral representations of three-point correlation functions and develop a method allowing us to compute them exactly. In particular, we evaluate the generalization of Selberg integral obtained by insertion of parafermionic polynomial. Our result is justified by different approach based on dual representation of parafermionic Liouville field theory described by three-exponential model.

hep-th↗