Abstract
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.
Highlights
Numerical conformal bootstrap is a very general and powerful approach to quantum conformal filed theories (CFTs), based on the idea of analyzing the crossing symmetry [1,2,3] of correlation functions in unitary CFTs by methods of semidefinite programming [4,5,6,7,8]
We study the structure of series expansions of general spinning conformal blocks
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
Summary
Numerical conformal bootstrap is a very general and powerful approach to quantum conformal filed theories (CFTs), based on the idea of analyzing the crossing symmetry [1,2,3] of correlation functions in unitary CFTs by methods of semidefinite programming [4,5,6,7,8]. Another approach to spinning conformal blocks is to relate them to simpler conformal blocks by means of differential operators [13, 78, 79]; recently it was shown that the most general conformal blocks can be reduced in this way to scalar blocks [80] While these methods do allow us to calculate any given non-supersymmetric conformal block, all of them currently require a nontrivial amount of case-specific analysis. In our form, the Casimir recursion relations can be immediately translated into a computer algorithm in all cases when the 6j symbols can be computed algorithmically This includes the general conformal blocks in 3 and 4 dimensions as well as seed blocks in higher dimensions.
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