Supergravity with Lagrange Multiplier Fields in 2 + 1 Dimensions

We examine the first-order Einstein-Cartan (EC) action in 2+1 dimensions, including a cosmological term and its supersymmetric extension. In this setting the spin connection can be expressed as an axial vector, yielding an action that is bilinear in the quantum fields and allows quantization without background fields. From the complete set of first-class constraints, the associated gauge transformations all follow. These differ from the standard supersymmetric diffeomorphism and local Lorentz invariances. Using the closed gauge algebra, we construct the Faddeev-Popov-Nielsen path integral and then show how a Lagrange multiplier field can be introduced to remove higher-loop contributions while preserving unitarity and gauge invariance. This indicates the possibility of being able to use a Lagrange multiplier field to formulate a unitary and renormalizable model for Einstein-Cartan gravity coupled to the Standard Model in 3+1 dimensions.