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

TL;DR

This paper introduces a new series representation for Virasoro fusion and modular kernels applicable at any irrational central charge, aligning with known formulas in specific cases and verified numerically.

Contribution

It provides a unified series representation for Virasoro kernels at irrational central charges, bridging different known formulas and offering numerical validation.

Findings

Series representation matches fusion transformation for c<1

Series representation agrees with Ponsot-Teschner formula for complex c

Numerical validation confirms the proposed formulas

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 $c\in \mathbb C \backslash (-\infty,1]$ 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 $c\in \mathbb C \backslash (-\infty,1]$ 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.