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

We propose a series representation for the Virasoro fusion and modular kernels at any irrational central charge. Two distinct, yet closely related formulas are needed for the cases $c\in \mathbb{C}\mathrm{\backslash }(-\mathrm{\infty },1]$ and $c<1$. Our proposal for $c<1$ agrees numerically with the fusion transformation of the four-point spherical conformal blocks, whereas our proposal for $c\in \mathbb{C}\mathrm{\backslash }(-\mathrm{\infty },1]$ agrees numerically with Ponsot and Teschner’s integral formula for the fusion kernel. The case of the modular kernel is studied as a special case of the fusion kernel.