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

21 pages

References in corpus (4)

• Differential equation for four-point correlation function in Liouville field theory and elliptic four-point conformal blocks
• S-duality as a beta-deformed Fourier transform
• Notes on crossing transformations of Virasoro conformal blocks
• On the Virasoro fusion kernel at $c=25$