Related papers: Bilinear equations on Painleve tau functions from …

Gamayun, Iorgov and Lisovyy in 2012 proposed that tau function of the Painlev\'e equation is equal to the series of $c=1$ Virasoro conformal blocks. We study similar series of $c=-2$ conformal blocks and relate it to Painlev\'e theory. The…

Iorgov, Lisovyy, and Teschner established a connection between isomonodromic deformation of linear differential equations and Liouville conformal field theory at $c=1$. In this paper we present a $q$ analog of their construction. We show…

Generic Painlev\'e VI tau function \tau(t) can be interpreted as four-point correlator of primary fields of arbitrary dimensions in 2D CFT with c=1. Using AGT combinatorial representation of conformal blocks and determining the…

We study explicit formula (suggested by Gamayun, Iorgov, Lisovyy) for Painlev\'e III($D_8$) $\tau$ function in terms of Virasoro conformal blocks with central charge $1$. The Painlev\'e equation has two types of bilinear forms, we call them…

We derive bilinear tau forms of the canonically quantized Painlev\'e equations, thereby relating them to those previously obtained from the $\mathbb{C}^2/\mathbb{Z}_2$ blowup relations for the $\mathcal{N}=2$ supersymmetric gauge theory…

We prove a Fredholm determinant and short-distance series representation of the Painlev\'e V tau function $\tau(t)$ associated to generic monodromy data. Using a relation of $\tau(t)$ to two different types of irregular $c=1$ Virasoro…

We study four dimensional supersymmetric gauge theory in the presence of surface and point-like defects (blowups) and propose an identity relating partition functions at different values of $\Omega$-deformation parameters…

The goal of this note is to show that the Riemann-Hilbert problem to find multivalued analytic functions with $SL(2,\mathbb{C})$-valued monodromy on Riemann surfaces of genus zero with $n$ punctures can be solved by taking suitable linear…

Painlev\`e equation for conformal blocks is a combined corollary of integrability and Ward identities, which can be explicitly revealed in the matrix model realization of AGT relations. We demonstrate this in some detail, both for…

We propose $q$-deformation of the Gamayun-Iorgov-Lisovyy formula for Painlev\'e $\tau$ function. Namely we propose formula for $\tau$ function for $q$-difference Painlev\'e equation corresponding to $A_7^{(1)}{}'$ surface (and $A_1^{(1)}$…

Generic c=1 four-point conformal blocks on the Riemann sphere can be seen as the coefficients of Fourier expansion of the tau function of Painlev\'e VI equation with respect to one of its integration constants. Based on this relation, we…

We elaborate on a recently conjectured relation of Painlev\'e transcendents and 2D CFT. General solutions of Painlev\'e VI, V and III are expressed in terms of $c=1$ conformal blocks and their irregular limits, AGT-related to instanton…

Classical conformal blocks naturally appear in the large central charge limit of 2D Virasoro conformal blocks. In the $AdS_{3}/CFT_{2}$ correspondence, they are related to classical bulk actions and are used to calculate entanglement…

This note details the relationship between the isomonodromic tau-function and conformal blocks, on a torus with one simple pole. It is based on the author's talk at ICMP 2021.

This paper is based on my presentation at RIMS workshop on "Theory of Integrable Systems and Its Applications in Various Fields" held in Kyoto on 19--21, August 2015. The aim of the present paper is to give a short account of recent studies…

In a recent paper (arXiv:0906.3219) the representation of Nekrasov partition function in terms of nontrivial two-dimensional conformal field theory has been suggested. For non-vanishing value of the deformation parameter…

We conjecture an expression for the Liouville theory conformal blocks and correlation functions on a Riemann surface of genus g and n punctures as the Nekrasov partition function of a certain class of N=2 SCFTs recently defined by one of…

By analogy to the continuous Painlev\'e II equation, we present particular solutions of the discrete Painlev\'e II (d-P$\rm_{II}$) equation. These solutions are of rational and special function (Airy) type. Our analysis is based on the…

We determine the spectrum and correlation functions of Liouville theory with a central charge less than (or equal) one. This completes the definition of Liouville theory for all complex values of the central charge. The spectrum is always…

In recent years, the Fourier series (Zak transform) structure of the Painlev\'e I tau function has emerged in multiple contexts. Its main building block admits several conjectural interpretations, such as the partition function of an…