On the Virasoro fusion kernel at $c=25$

Abstract

We find a formula for the Virasoro fusion kernel at c=25c=25, in terms of the connection coefficients of the Painlevé VI differential equation. Our formula agrees numerically with previously known integral representations of the kernel. The derivation of our formula relies on a duality c\to 26-cc→26−c that is obeyed by the shift equations for the fusion and modular kernels. We conjecture that for c<1c<1 the fusion and modular kernels are not smooth functions, but distributions.