Supersymmetric Spin Networks and Quantum Supergravity

TL;DR

This work extends loop quantum gravity to ${N}=1$ supergravity by constructing supersymmetric spin networks based on the supergroup ${Osp(1|2)}$, enabling a gauge-invariant, background-independent basis for non-perturbative quantization. It develops a diagrammatic calculus for ${Osp(1|2)}$ spin networks, including a graphical representation of edges and vertices, and shows how to evaluate these networks by decomposing into ordinary ${SU(2)}$ spin networks. A supersymmetric area operator with a discrete spectrum is derived, yielding eigenvalues $A = \sum_i \sqrt{ j_i(j_i+\tfrac{1}{2}) } l_p^2$, reflecting the $Osp(1|2)$ Casimir structure and extending the standard area spectrum of quantum gravity. The results suggest a framework for exploring ${N}=1$ supergravity non-perturbatively and indicate potential extensions to higher ${N}$, supersymmetric Yang–Mills theories, and spin-foam models with connections to string/M-theory.

Abstract

We define supersymmetric spin networks, which provide a complete set of gauge invariant states for supergravity and supersymmetric gauge theories. The particular case of Osp(1/2) is studied in detail and applied to the non-perturbative quantization of supergravity. The supersymmetric extension of the area operator is defined and partly diagonalized. The spectrum is discrete as in quantum general relativity, and the two cases could be distinguished by measurements of quantum geometry.

Figures (22)

• Figure 1: Recoupling theory for $Osp(1|2)$ 6-j symbols
• Figure 2: Unit element in fundamental representation of $Osp(1|2)$
• Figure 3: Construction of unit element in representation with spin one
• Figure 4: Decomposition of the link with color 2 in super spin networks into normal SU(2) ones
• Figure 5: Decomposition of the super link of color $n$