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We show that the conformal blocks constructed in the previous article by the first and the third author may be described as certain integrals in equivariant cohomology. When the bundles of conformal blocks have rank one, this construction may be compared with the old integral formulas of the second and the third author. The proportionality coefficients are some Selberg type integrals which are computed. Finally, a geometric construction of the tensor products of vector representations of the Lie algebra $frak{gl}(m)$ is proposed.
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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.
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.
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 review recent progress in operator algebraic approach to conformal quantum field theory. Our emphasis is on use of representation theory in classification theory. This is based on a series of joint works with R. Longo.
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