Analytic properties of the Virasoro modular kernel

TL;DR

This paper investigates the analytic properties of the Virasoro modular kernel, demonstrating that these properties, derived from Zamolodchikov's formula, are consistent with the kernel's behavior, supported by explicit calculations for one-point toric blocks.

Contribution

It establishes the shared analytic properties of the Virasoro modular kernel and conformal blocks, providing explicit computations for the one-point toric case.

Findings

Analytic properties of the modular kernel match those of conformal blocks.

Explicit computation confirms the theoretical predictions.

Supports the understanding of modular transformations in conformal field theory.

Abstract

On the space of generic conformal blocks the modular transformation of the underlying surface is realized as a linear integral transformation. We show that the analytic properties of conformal block implied by Zamolodchikov's formula are shared by the kernel of the modular transformation and illustrate this by explicit computation in the case of the one-point toric conformal block.