Towards a loop representation for quantum canonical supergravity

TL;DR

The paper develops a loop-quantized treatment of canonical supergravity by casting the theory in terms of a $GSU(2)$ connection and constructing a loop representation based on $GSU(2)$ Wilson loops with fermionic content. It identifies exact, gauge-invariant solutions to the quantum constraints, including a diffeomorphism-invariant Chern-Simons state, and shows how the loop transform of this state yields the Dubrovnik version of the Kauffman polynomial, with explicit skein relations tied to the theory's constraints. This establishes a direct link between a supersymmetric Chern-Simons state and a topological knot invariant, enabling potential analytic expressions for knot polynomial coefficients via perturbative CS methods. The work also outlines open challenges, such as fully implementing the left SUSY constraint in the loop framework and developing graded spin-network bases for the physical state space.

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.

Figures (5)

• Figure 1: The different crossings involved in the skein relations
• Figure 2: Regular isotopy invariants are not invariant under the addition and removal of a curl. This is determined by the skein relations involving the elements shown
• Figure 3: The addition of a small loop at an intersection
• Figure 4: The same "straight-through" crossing can correspond to a two loop link or a single loop depending on the connectivity of the diagram.
• Figure 5: Intersections without (a) and with (b) kinks. Under the present regularization intersections of type (b) are equivalent to no intersection (c).