$\mathcal{N}=1$ Supergravity with LQG methods and quantization of the SUSY constraint

TL;DR

This paper develops a loop quantum gravity framework for $ =1$ supergravity, deriving a quantum supersymmetry constraint operator and exploring its implications for the quantum theory's structure and physical states.

Contribution

It introduces a novel quantization of the supersymmetry constraint within LQG, providing explicit operator expressions and analyzing the algebraic structure of the quantum constraints.

Findings

Derived a compact expression for the SUSY constraint

Quantized the SUSY constraint operator explicitly

Discussed properties of solutions to the quantum SUSY constraint

Abstract

In this paper, the classical and quantum theory of $\mathcal{N}=1$ supergravity in four spacetime dimensions will be studied in the framework of loop quantum gravity. We discuss the canonical analysis of the supergravity Holst action as first introduced by Tsuda. In this way, we also derive a compact expression of the supersymmetry constraint, which plays a crucial role in canonical supergravity theories, akin to the role of the Hamiltonian constraint in non-supersymmetric generally covariant theories. The resulting theory is then quantized using loop quantum gravity methods. In particular, we propose and discuss a quantization of the supersymmetry constraint and derive explicit expressions of the action of the resulting operator. This is important as it is the first step on the way of analyzing the Dirac algebra generated by supersymmetry and Hamiltonian constraint in the quantum theory and for finding physical states. We also discuss some qualitative properties of such solutions of the SUSY constraint.