Torus conformal blocks and Casimir equations in the necklace channel · arXivDesk
2205.05038May 10, 202227 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
Torus conformal blocks and Casimir equations in the necklace channel
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
Nearby in the stack
regime, where the Virasoro algebra truncates to the
sl(2,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→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.