Abstract
We initiate the study of the conformal bootstrap using Sturm-Liouville theory, specializing to four-point functions in one-dimensional CFTs. We do so by decomposing conformal correlators using a basis of eigenfunctions of the Casimir which are labeled by a complex number α. This leads to a systematic method for computing conformal block decompositions. Analyzing bootstrap equations in alpha space turns crossing symmetry into an eigenvalue problem for an integral operator K. The operator K is closely related to the Wilson transform, and some of its eigenfunctions can be found in closed form.
Highlights
Since the bootstrap uses constraints coming from correlation functions, it is natural to express crossing symmetry as a sum rule in position space
In the present paper we introduce alpha space, an integral transform for CFT correlators based on the Sturm-Liouville theory of the conformal Casimir operator
Inspired by classic results [73], we discussed the decomposition of a CFT four-point correlator in terms of a new basis of functions Ψα(z) and explained how the familiar conformal block decomposition can be obtained by analytic continuation in α
Summary
This section is devoted to the Sturm-Liouville theory of the conformal Casimir of SL(2, R), the conformal group in one spacetime dimension. The function Fφφφφ(z) admits the following conformal block (CB). We notice that D can be written in the following suggestive form: D. where f, g are functions (0, 1) → C that are well-behaved near z = 0 and z = 1. Not all functions have a finite norm with respect to the inner product (2.7). Requiring that a function f is square integrable leads to the following constraints on its asymptotics near z = 0 and z = 1:. This implies that in a unitary CFT all four-point functions Fφφφφ(z) have a divergent norm with respect to (2.7). We can decompose a given function f : (0, 1) → R as follows: This formula describes how f (z) is encoded by its “spectral density” f (α), and vice versa.
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