Connection Problem for the Sine-Gordon/Painlevé III Tau Function and Irregular Conformal Blocks: Fig. 1.

Abstract

Abstract. The short-distance expansion of the tau function of the radial sine-Gordon/Painlevé III equation is given by a convergent series which involves irregular c = 1 conformal blocks and possesses certain periodicity properties with respect to monodromy data. The long-distance irregular expansion exhibits a similar periodicity with respect to a different pair of coordinates on the monodromy manifold. This observation is used to conjecture an exact expression for the connection constant providing relative normalization of the two series. Up to an elementary prefactor, it is given by the generating function of the canonical transformation between the two sets of coordinates.

The short pulse equation by a Riemann–Hilbert approach

“…In Section 3 we give a proof for the conjecture of [24] in the four-dimensional limit by using the well known results on spectral determinants [28] together with the recent developments in the context of Painlevé equations [18][19][20][21][22][23]. More precisely we show that both the l.h.s and the r.h.s of (1.2) satisfy the Painlevé III 3 equation in the τ form with the same initial conditions.…”

“…The solution (3.27) is a convergent, well-defined function whenever 2σ / ∈ Z [19,22]. Due to the periodicity properties it is enough to consider σ = 0, 1/2, for these values there exists a regularization procedure leading to a well-defined solution with η → 0 3 .…”

“…The first one is the connection between four-dimensional N = 2 gauge theories and Painlevé transcendants found in [18][19][20][21][22][23] and the other is a recently proposed conjecture relating topological strings and spectral theory [24].…”

“…The main result of [18][19][20][21][22][23] is that the Nekrasov-Okounkov (NO) partition function of SU (2) four-dimensional gauge theory calculates the τ function of a corresponding Painlevé equation.…”

Isomonodromic τ-functions and W N conformal blocks

“…This conjecture has already been proved in two ways: one proof is purely representation-theoretic and adapted initially for the 4-point τ -function [9] but can provide us with a collection of JHEP09 (2015)167 nontrivial bilinear relations for the n-point conformal blocks, whereas another one is based on the computation of monodromies of conformal blocks with degenerate fields and allows to consider an arbitrary number of regular singular points on the Riemann sphere [10]. The correspondence also extends to the irregular case: for instance, it gives exact solutions of the Painlevé V and III equations [11,12], which are known to describe correlation functions in certain massive field theories.…”