Crossing antisymmetric Polyakov blocks + Dispersion relation

TL;DR

This paper introduces '+ type Polyakov blocks' built from AdS Witten diagrams to analyze crossing antisymmetric sectors in CFT correlation functions, establishing a basis from dispersion relations and simplifying analytic functionals.

Contribution

It constructs a new basis of crossing antisymmetric objects from Witten diagrams, derived via dispersion relations, to facilitate analysis of antisymmetric sectors in CFTs.

Findings

Polyakov blocks encode crossing antisymmetric functions.

Dispersion relations impose locality constraints.

Simplifies analytic functionals in higher dimensions.

Abstract

Many CFT problems, e.g. ones with global symmetries, have correlation functions with a crossing antisymmetric sector. We show that such a crossing antisymmetric function can be expanded in terms of manifestly crossing antisymmetric objects, which we call the '+ type Polyakov blocks'. These blocks are built from AdS$_{d+1}$ Witten diagrams. In 1d they encode the '+ type' analytic functionals which act on crossing antisymmetric functions. In general d we establish this Witten diagram basis from a crossing antisymmetric dispersion relation in Mellin space. Analogous to the crossing symmetric case, the dispersion relation imposes a set of independent 'locality constraints' in addition to the usual CFT sum rules given by the 'Polyakov conditions'. We use the Polyakov blocks to simplify more general analytic functionals in $d > 1$ and global symmetry functionals.