On the Virasoro fusion kernel at $c=25$

Abstract

We find a formula for the Virasoro fusion kernel at $c=25$, 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 $c\to 26-c$ that is obeyed by the shift equations for the fusion and modular kernels. We conjecture that for $c<1$ the fusion and modular kernels are not smooth functions, but distributions.

18 pages, plus an ancillary Python notebook -- v2: clarified a few points, discussed the series representation

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Cited by in corpus (1)

• On the Virasoro fusion and modular kernels at any irrational central charge