Gravity and Matter from Soldered Chiral Superconnections

Abstract: We present a reformulation of four-dimensional chiral gravity in terms of a chiral gauge connection and a Clifford-valued coframe. The exterior algebra generated by $e^a γ_a$, together with the gauge curvature $F$, provides a natural and systematic construction of gauge-covariant differential forms. We show that suitable combinations of the chiral actions reproduce Einstein--Cartan--Holst gravity with a cosmological term. The original superconnection invariants yield only algebraic fermionic equations, so the fermionic one-forms do not propagate but instead generate a mass-like mixing between the two chiralities. Spinor propagation is supplied by a spinorial BF-type term, which under the matter ansatz reduces to the standard first-order Weyl kinetic action, including the torsion coupling. We also review the symmetry and variational properties of the matter ansatz, including its Lorentz equivariance, its relation to residual supersymmetry and twistor spinors, and the connection between the reduced and unrestricted field equations.