Painlevé VI connection problem and monodromy of c=1 conformal blocks
N. Iorgov, O. Lisovyy, Yu. Tykhyy
TL;DR
The work establishes a precise link between $c=1$ four-point Virasoro conformal blocks and the Painlevé VI tau function, identifying the $c=1$ fusion matrix with the Painlevé VI tau-function connection coefficient. By solving functional relations derived from tau-function expansions, the authors derive an explicit, nonperturbative formula for the connection coefficient in terms of Barnes $G$-functions, expressed through monodromy data and a geometric generating function equal to a complexified hyperbolic tetrahedron volume. This leads to a concrete expression for the $c=1$ fusion kernel, showing it coincides with the connection coefficient up to elementary prefactors, and is verified through algebraic Painlevé VI solutions and Ashkin–Teller (AT) special cases. The results illuminate deep connections between isomonodromic deformations, hyperbolic geometry, and CFT dualities (AGT), and they provide new tools for exploring the $c=1$ boundary of the Moore–Seiberg fusion structure.
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 Painlevé 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.