Towards a loop representation for quantum canonical supergravity

TL;DR

This paper develops a loop representation for quantum supergravity using Ashtekar variables, revealing that physical states are knot invariants related to the Kauffman Polynomial, thus advancing the understanding of quantum supergravity's state space.

Contribution

It introduces a loop representation for supergravity with a $GSU(2)$ connection and explicitly links physical states to the Dubrovnik version of the Kauffman Polynomial.

Findings

Physical states are knot invariants compatible with $GSU(2)$ Mandelstam identities.

An explicit solution to quantum constraints is given by the Dubrovnik Kauffman Polynomial.

The approach enables potential explicit analytic expressions for knot polynomial coefficients.

Abstract

We study several aspects of the canonical quantization of supergravity in terms of the Asthekar variables. We cast the theory in terms of a $GSU(2)$ connection and we introduce a loop representation. The solution space is similar to the loop representation of ordinary gravity, the main difference being the form of the Mandelstam identities. Physical states are in general given by knot invariants that are compatible with the $GSU(2)$ Mandelstam identities. There is an explicit solution to all the quantum constraint equations connected with the Chern-Simons form, which coincides exactly with the Dubrovnik version of the Kauffman Polynomial. This provides for the first time the possibility of finding explicit analytic expressions for the coefficients of that knot polynomial.