Towards Loop Quantum Supergravity (LQSG) I. Rarita-Schwinger Sector

TL;DR

This paper extends connection formulations of Lorentzian supergravity theories, including 11d and 4d N=8, to incorporate the Rarita-Schwinger field within a background independent loop quantum gravity framework, addressing key algebraic challenges.

Contribution

It develops a background independent representation of the Rarita-Schwinger field's algebra, enabling loop quantum gravity methods to be applied to supergravity theories.

Findings

Successfully extends connection formulation to supergravity with Rarita-Schwinger fields.

Provides a novel algebraic representation for Majorana fermions in this context.

Addresses the challenge of extending gauge groups to Spin(D+1) in supergravity.

Abstract

In our companion papers, we managed to derive a connection formulation of Lorentzian General Relativity in D+1 dimensions with compact gauge group SO(D+1) such that the connection is Poisson commuting, which implies that Loop Quantum Gravity quantisation methods apply. We also provided the coupling to standard matter. In this paper, we extend our methods to derive a connection formulation of a large class of Lorentzian signature Supergravity theories, in particular 11d SUGRA and 4d, N = 8 SUGRA, which was in fact the motivation to consider higher dimensions. Starting from a Hamiltonian formulation in the time gauge which yields a Spin(D) theory, a major challenge is to extend the internal gauge group to Spin(D+1) in presence of the Rarita-Schwinger field. This is non trivial because SUSY typically requires the Rarita-Schwinger field to be a Majorana fermion for the Lorentzian Clifford algebra and Majorana representations of the Clifford algebra are not available in the same spacetime dimension for both Lorentzian and Euclidean signature. We resolve the arising tension and provide a background independent representation of the non trivial Dirac antibracket *-algebra for the Majorana field which significantly differs from the analogous construction for Dirac fields already available in the literature.