Extend the 2+1D Einstein–Cartan supergravity framework to 3+1 dimensions

Extend the analysis of the 2+1-dimensional Einstein–Cartan supergravity model to four spacetime dimensions by identifying the complete gauge structure that arises from the canonical formulation of the 3+1-dimensional Einstein–Cartan theory and determining how these symmetries can be consistently implemented in the quantized theory.

Background

The paper develops the gauge structure and quantization of the Einstein–Cartan action in 2+1 dimensions, including a cosmological term and a gravitino, showing that the first-class constraints lead to a closed gauge algebra distinct from standard diffeomorphism and local Lorentz invariances. It constructs the Faddeev–Popov–Nielsen path integral, derives BRST transformations, and demonstrates how introducing a Lagrange multiplier field can restrict radiative corrections to one loop while maintaining gauge invariance and unitarity.

In contrast, the four-dimensional Einstein–Cartan theory poses major challenges: the canonical constraint structure is complex, and the corresponding gauge symmetries have not been fully characterized or implemented consistently in a quantum framework. The authors explicitly identify the extension of their 2+1D results to 3+1D as an open problem, highlighting the need to first determine the complete gauge structure from the canonical formulation and then establish its consistent quantization.