On the Virasoro fusion kernel at $c=25$

We find a formula for the Virasoro fusion kernel at c=25 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mrow> <mml:mi>c</mml:mi> <mml:mo>=</mml:mo> <mml:mn>25</mml:mn> </mml:mrow> </mml:math> , 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 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mrow> <mml:mi>c</mml:mi> <mml:mo>→</mml:mo> <mml:mn>26</mml:mn> <mml:mo>−</mml:mo> <mml:mi>c</mml:mi> </mml:mrow> </mml:math> that is obeyed by the shift equations for the fusion and modular kernels. We conjecture that for c&lt;1 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mrow> <mml:mi>c</mml:mi> <mml:mo>&lt;</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:math> the fusion and modular kernels are not smooth functions, but distributions.