On modular transformations of non-degenerate toric conformal blocks

TL;DR

This paper derives explicit difference equations for the toric modular kernel of non-degenerate Virasoro conformal blocks, providing a simple series expansion that aligns with known integral representations and offers a non-perturbative perspective.

Contribution

It introduces a new explicit series expansion for the toric modular kernel, derived from fusion algebra consistency, connecting it to established integral formulas.

Findings

Derived and solved difference equations for the modular kernel

Established equivalence with known integral representations

Provided a non-perturbative series expansion

Abstract

We derive and solve the difference equations on the toric modular kernel following from the consistency relations in the fusion algebra. The result is explicit and simple series expansion for the toric modular kernel of non-degenerate Virasoro conformal blocks. We show that this expansion is equivalent to the celebrated integral representation due to B. Ponsot and J. Teschner. We also interpret obtained series representation as a non-perturbative expansion and note that this raises further questions.