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 at infinity. 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.
Highlights
A decade after its modern renaissance [1], the conformal bootstrap program continues to undergo rapid development
We develop an analytic approach to the four-point crossing equation in CFT, for general spacetime dimension
We claim that s- and t-channel double-trace blocks and their derivatives with respect to ∆ form a basis for superbounded four-point functions
Summary
A decade after its modern renaissance [1], the conformal bootstrap program continues to undergo rapid development. We develop analytic functionals for four-point functions in a CFT in general dimension d > 1. While this is a priori a more complicated setup, the presence of two independent cross-ratios will allow for more flexible complex-analytic manipulations, and lead to somewhat simpler functionals than in the one-dimensional case. We claim that s- and t-channel double-trace blocks and their derivatives with respect to ∆ form a basis for superbounded four-point functions.. There are linear relations arising from the existence of bounded contact Witten diagrams, which can be separately expanded in either s- or the t-channel double-trace blocks and their derivatives.
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