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

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.