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, 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.
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Alkalaev, K., Mandrygin, S. & Pavlov, M. Torus conformal blocks and Casimir equations in the necklace channel. J. High Energ. Phys. 2022, 91 (2022). https://doi.org/10.1007/JHEP10(2022)091
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DOI: https://doi.org/10.1007/JHEP10(2022)091