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

Abstract

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∈ \mathbb{C} \backslash (-∞,1]c∈ℂ∖(−∞,1] and c <1c<1. Our proposal for c <1c<1 agrees numerically with the fusion transformation of the four-point spherical conformal blocks, whereas our proposal for c∈ \mathbb{C} \backslash (-∞,1]c∈ℂ∖(−∞,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.