  We find a formula for the Virasoro fusion kernel at $c=25$, in terms of the
connection coefficients of the Painlev\'e 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.
