Six-dimensional Methods for Four-dimensional Conformal Field Theories II: Irreducible Fields

TL;DR

This paper extends six-dimensional methods to construct four-dimensional conformal primary fields, focusing on irreducible representations involving the four-dimensional Levi-Civita tensor, enhancing the understanding of conformal field theories.

Contribution

It introduces constraints on six-dimensional fields to produce four-dimensional irreducible primary fields, including those involving the Levi-Civita tensor, advancing the construction of conformal fields.

Findings

Method to constrain six-dimensional fields for irreducible four-dimensional representations

Inclusion of Levi-Civita tensor in irreducibility conditions

Enhanced construction of conformal primary fields in four dimensions

Abstract

This note supplements an earlier paper on conformal field theories. There it was shown how to construct tensor, spinor, and spinor-tensor primary fields in four dimensions from their counterparts in six dimensions, where conformal transformations act simply as SO(4,2) Lorentz transformations. Here we show how to constrain fields in six dimensions so that the corresponding primary fields in four dimensions transform according to irreducible representations of the four-dimensional Lorentz group, even when the irreducibility conditions on these representations involve the four-component Levi-Civita tensor $\epsilon_{\mu\nu\rho\sigma}$.