Supergravity with Lagrange Multiplier Fields in 2 + 1 Dimensions

TL;DR

This work develops a background-free quantization of the $2+1$ Einstein–Cartan gravity with a cosmological term and its supersymmetric extension by exploiting the dualization of the spin connection to an axial vector, yielding a bilinear action $S_{\text{EC}3}$. The authors identify the complete set of first-class constraints, derive a closed gauge algebra (which requires $\Lambda = -\varkappa^{2}$) and implement Faddeev–Popov–Nielsen quantization, including BRST transformations and gauge-parameter dependence. A key result is the introduction of a Lagrange multiplier field to enforce the classical equations of motion, which confines quantum corrections to one-loop order while preserving unitarity and gauge invariance. The analysis provides a concrete lower-dimensional template for incorporating gravity with matter in a renormalizable, unitary framework and outlines prospects for extending these methods to $3+1$ dimensions, potentially informing unified approaches to fundamental interactions.

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.