Torus conformal blocks and Casimir equations in the necklace channel. (English) Zbl 1534.81129
Summary: 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 \(k = 1, \dots, n\). 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 \(q\rightarrow0\), 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.
MSC:
| 81T40 | Two-dimensional field theories, conformal field theories, etc. in quantum mechanics |
| 81R10 | Infinite-dimensional groups and algebras motivated by physics, including Virasoro, Kac-Moody, \(W\)-algebras and other current algebras and their representations |
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