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Casimir recursion relations for general conformal blocks. (English) Zbl 1387.81324

Summary: We study the structure of series expansions of general spinning conformal blocks. We find that the terms in these expansions are naturally expressed by means of special functions related to matrix elements of Spin(\(d\)) representations in Gelfand-Tsetlin basis, of which the Gegenbauer polynomials are a special case. We study the properties of these functions and explain how they can be computed in practice. We show how the Casimir equation in Dolan-Osborn coordinates leads to a simple one-step recursion relation for the coefficients of the series expansion of general spinning conformal block. The form of this recursion relation is determined by \(6j\) symbols of Spin(\(d-1\)). In particular, it can be written down in closed form in \(d=3\), \(d=4\), for seed blocks in general dimensions, or in any other situation when the required \(6j\) symbols can be computed. We work out several explicit examples and briefly discuss how our recursion relation can be used for efficient numerical computation of general conformal blocks.

MSC:

81T40 Two-dimensional field theories, conformal field theories, etc. in quantum mechanics
81T55 Casimir effect in quantum field theory
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