Torus conformal blocks and Casimir equations in the necklace channel

• Published in: The journal of high energy physics Vol. 2022; no. 10; pp. 91 - 26
• Main Authors: Alkalaev, Konstantin, Mandrygin, Semyon, Pavlov, Mikhail
• Format: Journal Article
• Language: English
• Published: Berlin/Heidelberg Springer Berlin Heidelberg 14.10.2022; Springer Nature B.V; SpringerOpen
• Subjects: Apexes; Channels; Classical and Quantum Gravitation; Closed loops; Conformal and W Symmetry; Critical phenomena; Decomposition; Elementary Particles; Field Theories in Lower Dimensions; Field theory; High energy physics; Mathematical analysis; Operators (mathematics); Physics; Physics and Astronomy; Quantum Field Theories; Quantum Field Theory; Quantum Physics; Regular Article - Theoretical Physics; Relativity Theory; String Theory; Symmetry; Toruses; Field Theories in Lower Dimensions; Conformal and W Symmetry
• ISSN: 1029-8479; 1029-8479
• DOI: 10.1007/JHEP10(2022)091

A bstract 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.