Towards Loop Quantum Supergravity (LQSG) II. p-Form Sector

TL;DR

This paper advances the quantisation of supergravity by extending loop quantum gravity methods to include bosonic p-form fields, exemplified through the 11-dimensional 3-index photon, revealing a consistent background-independent quantum structure.

Contribution

It introduces a novel quantisation approach for bosonic p-form fields in supergravity within the loop quantum gravity framework, including the effects of Chern-Simons interactions.

Findings

Successful phase space quantisation of the 3-index photon in 11d SUGRA.

Identification of a background-independent state for the Weyl algebra with Chern-Simons twist.

Extension of methods to general p-form fields in supergravity theories.

Abstract

In our companion paper, we focussed on the quantisation of the Rarita-Schwinger sector of Supergravity theories in various dimensions by using an extension of Loop Quantum Gravity to all spacetime dimensions. In this paper, we extend this analysis by considering the quantisation of additional bosonic fields necessary to obtain a complete SUSY multiplet next to graviton and gravitino in various dimensions. As a generic example, we study concretely the quantisation of the 3-index photon of 11d SUGRA, but our methods easily extend to more general p-form fields. Due to the presence of a Chern-Simons term for the 3-index photon, which is due to local SUSY, the theory is self-interacting and its quantisation far from straightforward. Nevertheless, we show that a reduced phase space quantisation with respect to the 3-index photon Gauss constraint is possible. Specifically, the Weyl algebra of observables, which deviates from the usual CCR Weyl algebras by an interesting twist contribution proportional to the level of the Chern-Simons theory, admits a background independent state of the Narnhofer-Thirring type.