Cauchy conformal fields in dimensions d>2

TL;DR

This paper extends the concept of holomorphic fields from 2d conformal field theory to higher dimensions by introducing Cauchy conformal fields, classifying unitary cases, and analyzing their properties and potential non-unitary extensions.

Contribution

It introduces and classifies Cauchy conformal fields in dimensions greater than two, generalizing holomorphic fields and analyzing their unitarity and correlation functions.

Findings

All unitary Cauchy fields are free and their correlation functions factorize.

Classification of unitary Cauchy fields in dimensions d>2.

Discussion of non-unitary Cauchy fields in d=3 and 4.

Abstract

Holomorphic fields play an important role in 2d conformal field theory. We generalize them to d>2 by introducing the notion of Cauchy conformal fields, which satisfy a first order differential equation such that they are determined everywhere once we know their value on a codimension 1 surface. We classify all the unitary Cauchy fields. By analyzing the mode expansion on the unit sphere, we show that all unitary Cauchy fields are free in the sense that their correlation functions factorize on the 2-point function. We also discuss the possibility of non-unitary Cauchy fields and classify them in d=3 and 4.