Torus conformal blocks and Casimir equations in the necklace channel

TL;DR

This paper develops a framework for understanding conformal blocks in two-dimensional conformal field theories on a torus, deriving Casimir equations in various channels and regimes, especially at large central charge and in the plane limit.

Contribution

It introduces a method to obtain Casimir equations for torus conformal blocks in arbitrary exchange channels, extending the understanding of their structure and computation.

Findings

Derived Casimir equations for torus conformal blocks in the necklace channel.

Explicitly formulated Casimir equations on the plane limit for global conformal blocks.

Proposed a general scheme to find Casimir equations in any channel.

Abstract

We consider the conformal block decomposition in arbitrary exchange channels of a two-dimensional conformal field theory on a torus. The channels are described by diagrams built of a closed loop with external legs (a necklace sub-diagram) and trivalent vertices forming trivalent trees attached to the necklace. Then, the $n$-point torus conformal block in any channel can be obtained by acting with a number of OPE operators on the $k$-point torus block in the necklace channel at $k=1,...,n$. Focusing on the necklace channel, we go to the large-$c$ regime, where the Virasoro algebra truncates to the $sl(2, \mathbb{R})$ subalgebra, and obtain the system of the Casimir equations for the respective $k$-point global conformal block. In the plane limit, when the torus modular parameter $q\to 0$, we explicitly find the Casimir equations on a plane which define the $(k+2)$-point global conformal block in the comb channel. Finally, we formulate the general scheme to find Casimir equations for global torus blocks in arbitrary channels.