Towards a loop representation for quantum canonical supergravity

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.

12 pages Revtex, figures with epsf. An important revision was made of section four. We were able to identify the exact quantum state of all the constraints of supergravity as the Dubrovnik version of the Kauffman polynomial. This provides the first known link of that polynomial to quantum field theory

References in corpus (11)

• Jones Polynomials for Intersecting Knots as Physical States of Quantum Gravity
• Fermions in Quantum Gravity
• Recent Developments in Non-Perturbative Quantum Gravity
• How the Jones Polynomial Gives Rise to Physical States of Quantum General Relativity
• The Non-perturbative Canonical Quantization of the N=1 Supergravity
• About Loop States in Supergravity
• About a Super-Ashtekar-Renteln Ansatz
• Quantum Einstein-Maxwell Fields: A Unified Viewpoint from the Loop Representation
• Exact quantum state for N=1 supergravity
• The Relation between the Mandelstam and the Cayley-Hamilton Identities
• An extension of the Cayley-Hamilton theorem to the case of supermatrices

Cited by in corpus (22)

• Quantum geometry with intrinsic local causality
• Supersymmetric Spin Networks and Quantum Supergravity
• Ashtekar's formulation for $N=1,2$ supergravities as "constrained" BF theories
• Holographic Formulation of Quantum Supergravity
• An elementary introduction to loop quantum gravity
• Eleven dimensional supergravity as a constrained topological field theory
• Towards Loop Quantum Supergravity (LQSG) I. Rarita-Schwinger Sector
• Holst-MacDowell-Mansouri action for (extended) supergravity with boundaries and super Chern-Simons theory
• A candidate for a background independent formulation of M theory
• Quaternionic Quantization Principle in General Relativity and Supergravity
• Combinatorial dynamics in quantum gravity
• Towards black hole entropy in chiral loop quantum supergravity
• Supersymmetric minisuperspace models in self-dual loop quantum cosmology
• Super Cartan geometry and the super Ashtekar connection
• Canonical formulation of N = 2 supergravity in terms of the Ashtekar variable
• Towards a loop representation of connection theories defined over a super Lie algebra
• $\mathcal{N}=1$ Supergravity with LQG methods and quantization of the SUSY constraint
• Towards a background independent approach to M theory
• Unified model of loop quantum gravity and matter
• Introduction to supersymmetric spin networks
• Supergroup BF action for supergravity
• Hamiltonian Theory: generalizations to higher dimensions, supersymmetry and modified gravity