A note on pure connection formalism for unimodular gravity and its possible generalisations

TL;DR

The paper develops a pure-connection formulation of unimodular gravity in the Henneaux-Teitelboim framework and shows how a Plebanski-type action can be reduced to a pure connection action. It then introduces a broad class of deformations of UG defined by homogeneous defining functions, proving that all such theories propagate two complex degrees of freedom and have a closed constraint algebra. By analyzing the 3+1 decomposition, the work clarifies the role of the unimodular 3-form and how a cosmological-constant-like global DoF emerges, while leaving reality conditions for future work. The results provide a framework for investigating perturbative quantum dynamics and potential connections to generalized UG formulations, with reality conditions and real-section extraction as key future directions.

Abstract

In this note, we consider the Henneaux-Teitelboim version of Unimodular Gravity (UG) and its deformations in the form of gauge theories with spontaneously broken diffeomorphism invariance. Actions defining such theories depends on the curvature of an $SO(3,\mathbb{C})$ gauge connection and the field strength of a (real) 3-form (or equivalently its dual vector density). We obtain the pure connection action of the theory from the corresponding Plebanski action by integrating out auxiliary fields. Then we show that the Henneaux-Teitelboim form of UG can be included in a wider class of theories which propagate two (complex) degrees of freedom.