Five-Dimensional Path Integrals for Six-Dimensional Conformal Field Theories

TL;DR

This paper develops a method to derive six-dimensional conformal field theory correlators from five-dimensional supersymmetric path integrals with instantons, revealing non-perturbative symmetry enhancements and quantized couplings.

Contribution

It introduces a reverse construction of six-dimensional correlators from five-dimensional theories, incorporating instantons and symmetry enhancements, enabling new computational approaches.

Findings

Ward-Takahashi identities derived for 5D theories with $SU(1,3)$ symmetry.

Correlation functions constructed from 5D operators satisfy 6D conformal Ward identities.

Coupling constants become quantized when instantons are included.

Abstract

In this paper we derive Ward-Takahashi identities from the path integral of supersymmetric five-dimensional field theories with an $SU(1,3)$ spacetime symmetry in the presence of instantons. We explicitly show how $SU(1,3)$ is enhanced to $SU(1,3)\times U(1)$ where the additional $U(1)$ acts non-perturbatively. Solutions to such Ward-Takahashi identities were previously obtained from correlators of six-dimensional Lorentzian conformal field theories but where the instanton number was replaced by the momentum along a null direction. Here we study the reverse procedure whereby we construct correlation functions out of towers of five-dimensional operators which satisfy the Ward-Takahashi identities of a six-dimensional conformal field theory. This paves the way to computing observables in six dimensions using five-dimensional path integral techniques. We also argue that, once the instanton sector is included into the path integral, the coupling of the five-dimensional Lagrangian must be quantised, leaving no free continuous parameters.