Torus conformal blocks and Casimir equations in the necklace 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.

arXiv comment27 pages, v2: a new section on Casimir equations in general torus channels, a new appendix containing explicit expressions for lower-point global torus blocks, minor edits, typos corrected, more refs added; v3: extended discussion of the conformal block decomposition in torus CFT2, more clarifying comments in the introduction, notations improved, typos corrected, journal version