Supergravity with Lagrange Multiplier Fields in 2 + 1 Dimensions

TL;DR

This paper explores a first-order supergravity formulation in 2+1 dimensions, introducing Lagrange multiplier fields to simplify quantization and maintain gauge invariance, with implications for quantum gravity models.

Contribution

It introduces a novel use of Lagrange multiplier fields in 2+1 dimensional supergravity to facilitate quantization and control higher-loop contributions.

Findings

Derived the complete set of first-class constraints.

Constructed the gauge algebra and path integral.

Showed how Lagrange multipliers preserve unitarity and gauge invariance.

Abstract

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. We identify the complete set of first-class constraints and derive the associated gauge transformations, which differ from the standard diffeomorphism and local Lorentz invariances. Using the closed gauge algebra, we construct the Faddeev-Popov-Nielsen path integral and show how a Lagrange multiplier field can be introduced to remove higher-loop contributions while preserving unitarity and gauge invariance.