Unimodular Pleba\'{n}ski Gravity

TL;DR

This paper introduces new action principles for unimodular gravity within the chiral Plebański formulation, allowing the cosmological constant to emerge as an integration constant and exploring their reduction to a pure connection form.

Contribution

It develops novel unimodular gravity actions in the Plebański framework and connects them to Krasnov's pure connection formalism, highlighting differences from metric-based approaches.

Findings

Cosmological constant appears as an integration constant.

Theories can be reduced to a pure connection form.

Discussion of Lorentzian solutions in complex theory.

Abstract

We present new action principles for unimodular gravity, defined in the chiral Pleba\'{n}ski formulation based on (complex) two-forms and a complex ${\rm SO}(3)$ connection. In these theories, just as in their analogues in the metric formulation, the cosmological constant does not take a prescribed value but is an integration constant whose value can differ between different (classical) solutions. We discuss some subtleties when identifying Lorentzian solutions in the generally complex theory, and show how these theories can be reduced to a ``pure connection'' form similar to Krasnov's pure connection formalism for general relativity.