Painlev\'e VI connection problem and monodromy of c=1 conformal blocks

TL;DR

This paper explores the deep connection between Painlevé VI equations and c=1 conformal blocks, providing explicit formulas for fusion and connection coefficients linked to hyperbolic geometry.

Contribution

It establishes a direct relation between c=1 fusion matrices and tau function connection coefficients, deriving explicit Barnes G-function formulas without integration.

Findings

Explicit formulas for connection coefficients involving Barnes G-functions.

Relation between conformal blocks and hyperbolic tetrahedron volume.

Fusion matrix coincides with tau function connection coefficient.

Abstract

Generic c=1 four-point conformal blocks on the Riemann sphere can be seen as the coefficients of Fourier expansion of the tau function of Painlev\'e VI equation with respect to one of its integration constants. Based on this relation, we show that c=1 fusion matrix essentially coincides with the connection coefficient relating tau function asymptotics at different critical points. Explicit formulas for both quantities are obtained by solving certain functional relations which follow from the tau function expansions. The final result does not involve integration and is given by a ratio of two products of Barnes G-functions with arguments expressed in terms of conformal dimensions/monodromy data. It turns out to be closely related to the volume of hyperbolic tetrahedron.