Unimodular Plebański Gravity

We present new action principles for unimodular gravity, defined in the chiral Plebański formulation based on (complex) two-forms and a complex $\mathrm{S}\mathrm{O}(3)$ 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.