Conformal geometry and (super)conformal higher-spin gauge theories

Abstract

We develop a manifestly conformal approach to describe linearised (super)conformal higher-spin gauge theories in arbitrary conformally flat backgrounds in three and four spacetime dimensions. Closed-form expressions in terms of gauge prepotentials are given for gauge-invariant higher-spin (super) Cotton and (super) Weyl tensors in three and four dimensions, respectively. The higher-spin (super) Weyl tensors are shown to be conformal primary (super)fields in arbitrary conformal (super)gravity backgrounds, however they are gauge invariant only if the background (super) Weyl tensor vanishes. The proposed higherspin actions are (super) Weyl-invariant on arbitrary curved backgrounds, however the appropriate higher-spin gauge invariance holds only in the conformally flat case. We also describe conformal models for generalised gauge fields that are used to describe partially massless dynamics in three and four dimensions. In particular, generalised higher-spin Cotton and Weyl tensors are introduced.

Generalised conformal higher-spin fields in curved backgrounds

“…We begin by reviewing the gauge-invariant model for the maximal depth spin-2 field (corresponding to m = n = t = 2) in a Bach-flat background [15]. Upon degauging, its action coincides with the one studied earlier in Ref.…”

“…Throughout this work we make use of the conformal calculus described in [15] (building on the earlier work [31]), to where we refer the reader for further details. The parts of this formalism that are essential to the models constructed in this paper are as follows.…”

“…Models for generalised conformal higher-spin gauge fields [16][17][18][19][20][21][22][23][24][25] in arbitrary conformally flat backgrounds were first constructed in [15]. Below we summarise their main properties.…”

“…For instance, since the 1985 work by Fradkin and Tseytlin [1] it was reasonably clear that there should exist a consistent formulation for all CHS models on arbitrary conformally flat backgrounds. However, such a formulation has been developed only recently [15], and it also works for generalised CHS models. The latter describe conformal spin-s fields of depth t > 1, which have been studied for more than thirty years [16][17][18][19][20][21][22][23][24][25].…”

Symmetries of supergravity backgrounds and supersymmetric field theory

“…Constraints (4.3) naturally occur in the framework of conformal higher-spin gauge supermultiplets [58,59]. For m ≥ n > 0 such a supermultiplet is described by an unconstrained primary prepotential Υ α(m)α(n) defined modulo gauge transformations…”