A Basis of Analytic Functionals for CFTs in General Dimension

TL;DR

This paper introduces a new analytic basis for conformal bootstrap equations in general dimensions, utilizing double-twist conformal blocks, and provides algorithms to construct dual functionals, advancing the understanding of crossing symmetry in CFTs.

Contribution

It develops a basis of analytic functionals for the crossing equations in CFTs across all dimensions, with explicit construction algorithms and connections to dispersion relations.

Findings

Constructed a basis of conformal blocks for the crossing equation.

Developed two algorithms to build dual linear functionals.

Linked the basis to recent CFT dispersion relations.

Abstract

We develop an analytic approach to the four-point crossing equation in CFT, for general spacetime dimension. In a unitary CFT, the crossing equation (for, say, the s- and t-channel expansions) can be thought of as a vector equation in an infinite-dimensional space of complex analytic functions in two variables, which satisfy a boundedness condition in the u-channel Regge limit. We identify a useful basis for this space of functions, consisting of the set of s- and t-channel conformal blocks of double-twist operators in mean field theory. We describe two independent algorithms to construct the dual basis of linear functionals, and work out explicitly many examples. Our basis of functionals appears to be closely related to the CFT dispersion relation recently derived by Carmi and Caron-Huot.