Abstract. We elaborate on a recently conjectured relation of Painlevé transcendents and 2D CFT. General solutions of Painlevé VI, V and III are expressed in terms of c = 1 conformal blocks and their irregular limits, AGT-related to instanton partition functions in N = 2 supersymmetric gauge theories with N f = 0, 1, 2, 3, 4. Resulting combinatorial series representations of Painlevé functions provide an efficient tool for their numerical computation at finite values of the argument. The series involve sums over bipartitions which in the simplest cases coincide with Gessel expansions of certain Toeplitz determinants. Considered applications include Fredholm determinants of classical integrable kernels, scaled gap probability in the bulk of the GUE, and all-order conformal perturbation theory expansions of correlation functions in the sine-Gordon field theory at the free-fermion point.
Generic Painlevé VI tau function τ (t) can be interpreted as four-point correlator of primary fields of arbitrary dimensions in 2D CFT with c = 1. Using AGT combinatorial representation of conformal blocks and determining the corresponding structure constants, we obtain full and completely explicit expansion of τ (t) near the singular points. After a check of this expansion, we discuss examples of conformal blocks arising from Riccati, Picard, Chazy and algebraic solutions of Painlevé VI.
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