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Bilinear equations on Painleve tau functions from CFT

M. A. Bershtein, A. I. Shchechkin

TL;DR

The paper proves that the Painlevé VI τ-function can be written as an infinite sum over $c=1$ Virasoro conformal blocks by establishing bilinear relations on conformal blocks derived from the embedding $ extsf{Vir}oxplus extsf{Vir} o extsf{F}oxplus extsf{NSR}$ and linking to AGT/Nekrasov counts on the resolution $X_2$. It provides explicit blow-up factors $l_n(P,b)$ and, using a Whittaker/decomposition framework, derives bilinear identities that yield and generalize Painlevé III′$_3$ and Painlevé VI τ-function equations, while offering an efficient recursive algorithm for conformal-block computations. The results give a CFT–instanton counting bridge for isomonodromic problems and suggest extensions to $c eq 1$ and to $W_N$ algebras, with broader implications for exact τ-functions and their geometric interpretations. Overall, the work solidifies a deep link between Painlevé transcendents, Virasoro and NSR/CFT structures, and Nekrasov partition functions within the AGT framework.

Abstract

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).

Bilinear equations on Painleve tau functions from CFT

TL;DR

The paper proves that the Painlevé VI τ-function can be written as an infinite sum over Virasoro conformal blocks by establishing bilinear relations on conformal blocks derived from the embedding and linking to AGT/Nekrasov counts on the resolution . It provides explicit blow-up factors and, using a Whittaker/decomposition framework, derives bilinear identities that yield and generalize Painlevé III′ and Painlevé VI τ-function equations, while offering an efficient recursive algorithm for conformal-block computations. The results give a CFT–instanton counting bridge for isomonodromic problems and suggest extensions to and to algebras, with broader implications for exact τ-functions and their geometric interpretations. Overall, the work solidifies a deep link between Painlevé transcendents, Virasoro and NSR/CFT structures, and Nekrasov partition functions within the AGT framework.

Abstract

In 2012 Gamayun, Iorgov, Lisovyy conjectured an explicit expression for the Painlevé VI ~function in terms of the Liouville conformal blocks with central charge . We prove that proposed expression satisfies Painlevé VI ~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 (similarly to Nakajima-Yoshioka blow-up equations).

Paper Structure

This paper contains 17 sections, 10 theorems, 196 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Algebras and representations
  3. Verma modules
  4. Vacuum module
  5. Vertex operators and conformal blocks
  6. Conformal blocks and chain vectors
  7. $\textsf{Vir} \oplus \textsf{Vir}$ decomposition of chain vectors and vertex operators
  8. Matrix elements $\langle P,n |\Phi_{\alpha}(1) |P',n'\rangle$
  9. Bilinear relations on conformal blocks
  10. Painlevé equations and isomonodromic problem
  11. Proof of the Painlevé III$'_3$ $\tau$ function conjecture
  12. Proof of the Painlevé VI $\tau$ function conjecture
  13. Effective algorithm for conformal blocks calculation
  14. AGT relation
  15. Concluding remarks
  16. ...and 2 more sections

Key Result

Theorem 1.1

The expansion of the Painlevé $III'_3$$\tau$ function near $t=0$ can be written as

Figures (2)

  • Figure 1: Decomposition of $\pi^{\Delta^{{\textsf{NS}}}}_{\textsf{F}\oplus \textsf{NSR}}$ into direct sum of representations of the algebra $\mathsf{Vir}\oplus\mathsf{Vir}$. Each interior angle corresponds to the Verma module $\pi^n_{\textsf{Vir}\oplus \textsf{Vir}}$.
  • Figure 2: Diagram representing conformal block as a matrix element.

Theorems & Definitions (30)

  • Theorem 1.1
  • Proposition 2.1
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Theorem 2.1
  • proof
  • Remark 3.1
  • Proposition 3.1
  • ...and 20 more