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Register a new userConnection problem for the sine-Gordon/Painleve III tau function and irregular conformal blocks
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The short-distance expansion of the tau function of the radial sine-Gordon/Painleve 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.
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We study the relation of irregular conformal blocks with the Painleve III$_3$ equation. The functional representation for the quasiclassical irregular block is shown to be consistent with the BPZ equations of conformal field theory and the Hamilton-Jacobi approach to Painleve III$_3$. It leads immediately to a limiting case of the blow-up equations for dual Nekrasov partition function of 4d pure supersymmetric gauge theory, which can be even treated as a defining system of equations for both $c=1$ and $ctoinfty$ conformal blocks. We extend this analysis to the domain of strong-coupling regime where original definition of conformal blocks and Nekrasov functions is not known and apply the results to spectral problem of the Matheiu equations. Finally, we propose a construction of irregular conformal blocks in the strong coupling region by quantization of Painleve III$_3$ equation, and obtain in this way a general expression, reproducing $c=1$ and quasiclassical $ctoinfty$ results as its particular cases. We have also found explicit integral representations for $c=1$ and $c=-2$ irregular blocks at infinity for some special points.
We prove a Fredholm determinant and short-distance series representation of the Painleve 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 conformal blocks and the confluence from Painleve VI equation, connection formulas between the parameters of asymptotic expansions at $0$ and $iinfty$ are conjectured. Explicit evaluations of the connection constants relating the tau function asymptotics as $tto 0,+infty,iinfty$ are obtained. We also show that irregular conformal blocks of rank 1, for arbitrary central charge, are obtained as confluent limits of the regular conformal blocks.
Classical Virasoro conformal blocks are believed to be directly related to accessory parameters of Floquet type in the Heun equation and some of its conflue
The goal of this article is to rederive the connection between the Painleve $5$ integrable system and the universal eigenvalues correlation functions of double-scaled hermitian matrix models, through the topological recursion method. More specifically we prove, textbf{to all orders}, that the WKB asymptotic expansions of the $tau$-function as well as of determinantal formulas arising from the Painleve $5$ Lax pair are identical to the large $N$ double scaling asymptotic expansions of the partition function and correlation functions of any hermitian matrix model around a regular point in the bulk. In other words, we rederive the sine-law universal bulk asymptotic of large random matrices and provide an alternative perturbative proof of universality in the bulk with only algebraic methods. Eventually we exhibit the first orders of the series expansion up to $O(N^{-5})$.
The Brezin-Gross-Witten (BGW) model is one of the basic examples in the class of non-eigenvalue unitary matrix models. The generalized BGW tau-function $tau_N$ was constructed from a one parametric deformation of the original BGW model using the generalized Kontsevich model representation. It is a tau-function of the KdV hierarchy for any value of $Ninmathbb C$, where the case $N=0$ reduces to the original BGW tau-function. In this paper, we present a bosonic representation of $tau_N$ in terms of the $W_{1+infty}$ operators that preserves the KP integrability. This allows us to establish a connection between the (generalized) BGW and Kontsevich-Witten tau-functions using $GL(infty)$ operators, both considered as the basic building blocks in the theory of matrix models and partition functions.
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