Conformal Hypergeometry and Integrability

TL;DR

This paper explores the mathematical structures underlying conformal field theories, highlighting connections with hypergeometry, group theory, and integrable models, and providing an accessible introduction to the conformal bootstrap approach.

Contribution

It introduces the mathematical aspects of conformal partial wave expansions and their deep relations with hypergeometry and integrable models, offering new insights into conformal field theory.

Findings

Connections between conformal partial waves and hypergeometric functions.

Relations of conformal bootstrap with integrable models like Gaudin and Calogero-Sutherland.

Emphasis on mathematical structures underlying conformal field theories.

Abstract

Conformal field theories play a central role in modern theoretical physics with many applications to the understanding of phase transitions, gauge theories and even the quantum physics of gravity, through Maldacena's celebrated holographic duality. The key analytic tool in the field is the so-called conformal partial wave expansion, i.e. a Fourier-like decomposition of physical quantities into a basis of partial waves for the conformal group SO(1,d+1). This text provides an non-technical introduction to conformal field theory and the conformal bootstrap program with some focus on the mathematical aspects of conformal partial wave expansions. It emphasises profound relations with modern hypergeometry, group theory and integrable models of Gaudin and Calogero-Sutherland type.