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Register a new userAccessory parameters in confluent Heun equations and classical irregular conformal blocks
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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
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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.
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.
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.
It is shown that the Confluent Heun Equation (CHEq) reduces for certain conditions of the parameters to a particular class of Quasi-Exactly Solvable models, associated with the Lie algebra $sl (2,{mathbb R})$. As a consequence it is possible to find a set of polynomial solutions of this quasi-exactly solvable version of the CHEq. These finite solutions encompass previously known polynomial solutions of the Generalized Spheroidal Equation, Razavy Eq., Whittaker-Hill Eq., etc. The analysis is applied to obtain and describe special eigen-functions of the quantum Hamiltonian of two fixed Coulombian centers in two and three dimensions.
We give a simple iterative procedure to compute the classical conformal blocks on the sphere to all order in the modulus.
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