Torus classical conformal blocks

TL;DR

This paper extends the computation of classical conformal blocks from sphere topology to torus topology, deriving explicit results up to third order and comparing them with quantum limits, advancing understanding in conformal field theory.

Contribution

It introduces a method to compute classical conformal blocks on the torus, extending previous sphere-based techniques, and provides explicit third-order results.

Findings

Derived classical Ward identity for torus modulus variation

Mapped sphere four-source problem to torus topology

Computed classical blocks up to third order and compared with quantum limits

Abstract

After deriving the classical Ward identity for the variation of the action under a change of the modulus of the torus we map the problem of the sphere with four sources to the torus. We extend the method previously developed for computing the classical conformal blocks for the sphere topology to the tours topology. We give the explicit results for the classical blocks up to the third order in the nome included and compare them with the classical limit of the quantum conformal blocks. The extension to higher orders is straightforward.