Secondary Fields in D>2 Conformal Theories

TL;DR

This paper explores secondary fields generated by non-abelian currents and energy-momentum tensors in D≥3 conformal theories, revealing their structure, existence conditions, and relations to anomalous operator contributions, aiming to develop exact solutions.

Contribution

It introduces a framework for secondary fields in D≥3 conformal theories, linking their existence to anomalous contributions and deriving their properties and relations to fundamental operators.

Findings

Existence of secondary fields governed by anomalous operator contributions.

Derived coupling constants for scalar fields R_j and R_T.

Established closed equations for Green functions of secondary fields.

Abstract

We consider the secondary fields in $D$-dimensional space, $D\ge3$, generated by the non-abelian current and energy-momentum tensor. These fields appear in the operator product expansions $j^{a}_\mu(x)\phi(0)$ and $T_{\mu\nu}(x)\phi(0)$. The secondary fields underlie the construction proposed herein (see [1,2] for more details) and aimed at the derivation of exact solutions of conformal models in $D\ge3$. In the case of D=2 this construction leads to the known [5] exactly solvable models based on the infinite-dimensional conformal symmetry. It is shown that for $D\ge3$ the existence of the secondary fields is governed by the existence of anomalous operator contributions (the scalar fields $R_j$ and $R_T$ of dimensions $d_j = d_T = D-2$) into the operator product expansions $j^{a}_\mu j^{b}_\nu$ and $T_{\mu\nu} T_{\rho\sigma}$. The coupling constant between the field $R_j$ and the fundamental field is found. The fields $R_j$ and $R_T$ are shown to beget two infinite sets of secondary tensor fields of canonical dimensions $D-2+s$, where $s$ is the tensor rank. The current and the energy-momentum tensor belong to those families, their Green functions being expressed through the Green functions of the fields $R_j$ and $R_T$ correspondingly. We demonstrate that the Ward identities give rise to the closed set of equations for the Green functions of the fields $R_j$ and $R_T$.