The Gauge Theory of Weyl Group and its Interpretation as Weyl Quadratic Gravity

TL;DR

This paper develops a gauge theory framework for Weyl quadratic gravity, revealing new dualities involving torsion and non-metricity, and extends the gauge construction to Poincaré and conformal groups.

Contribution

It introduces a generalized duality between torsion and non-metricity, and shows how Weyl quadratic gravity encompasses gauge theories of Poincaré and conformal groups.

Findings

Weyl quadratic gravity can be formulated as a gauge theory of the Weyl group.

A new duality involving a traceless 3-tensor extends the torsion/non-metricity equivalence.

The gauge theory of the conformal group is a special case of Weyl quadratic gravity.

Abstract

In this paper we give an extensive description of Weyl quadratic gravity as the gauge theory of the Weyl group. The previously discovered (vectorial) torsion/non-metricity equivalence is shown to be built-in as it corresponds to a redefinition of the generators of the Weyl group. We present a generalisation of the torsion/non-metricity duality which includes, aside from the vector, also a traceless 3-tensor with two antisymmetric indices and vanishing skew symmetric part. A discussion of this relation in the case of minimally coupled matter fields is given. We further point out that a Rarita-Schwinger field can couple minimally to all the components of torsion and some components of non-metricity. Alongside we present the same gauge construction for the Poincar\'e and conformal groups. We show that even though the Weyl group is a subgroup of the conformal group, the gauge theory of the latter is actually only a special case of Weyl quadratic gravity.