Canonical analysis of unimodular Plebański gravity

We present the canonical analysis of different versions of unimodular gravity defined in the Plebański formalism, based on a (generally complex) <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"><a:mi>S</a:mi><a:mi>O</a:mi><a:mo stretchy="false">(</a:mo><a:mn>3</a:mn><a:mo stretchy="false">)</a:mo></a:math> spin connection and set of (self-dual) two-forms. As in the metric formulation of unimodular gravity, one can study either a theory with fixed volume form or work in a parametrized formalism in which the cosmological constant becomes a dynamical field, constrained to be constant by the field equations. In the first case, the Hamiltonian density contains a part which is not constrained to vanish, but rather constrained to be constant, again as in the metric formulation. We also discuss reality conditions and challenges in extracting Lorentzian solutions.