Towards Loop Quantum Supergravity (LQSG) I. Rarita-Schwinger Sector

Abstract

In our companion papers, we managed to derive a connection formulation of Lorentzian General Relativity in D+1 dimensions with compact gauge group SO(D+1) such that the connection is Poisson commuting, which implies that Loop Quantum Gravity quantisation methods apply. We also provided the coupling to standard matter. In this paper, we extend our methods to derive a connection formulation of a large class of Lorentzian signature Supergravity theories, in particular 11d SUGRA and 4d, N = 8 SUGRA, which was in fact the motivation to consider higher dimensions. Starting from a Hamiltonian formulation in the time gauge which yields a Spin(D) theory, a major challenge is to extend the internal gauge group to Spin(D+1) in presence of the Rarita-Schwinger field. This is non trivial because SUSY typically requires the Rarita-Schwinger field to be a Majorana fermion for the Lorentzian Clifford algebra and Majorana representations of the Clifford algebra are not available in the same spacetime dimension for both Lorentzian and Euclidean signature. We resolve the arising tension and provide a background independent representation of the non trivial Dirac antibracket *-algebra for the Majorana field which significantly differs from the analogous construction for Dirac fields already available in the literature.

43 pages. v2: Journal version. Some nonessential sign errors in sections 2 and 3 corrected. Minor clarifications and corrections

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• A new spinfoam vertex for quantum gravity
• The loop-quantum-gravity vertex-amplitude
• Flipped spinfoam vertex and loop gravity
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• Perelomov type coherent states of SO(D + 1) in all dimensional loop quantum gravity
• The Wald entropy formula and loop quantum gravity
• Superposition type coherent states in all dimensional loop quantum gravity
• Twisted geometry coherent states in all dimensional loop quantum gravity: I. Construction and Peakedness properties
• Towards black hole entropy in chiral loop quantum supergravity
• Connection Dynamics for Higher Dimensional Scalar-Tensor Theories of Gravity
• Twisted geometry coherent states in all dimensional loop quantum gravity: II. Ehrenfest Property
• Super Cartan geometry and the super Ashtekar connection
• $\mathcal{N}=1$ Supergravity with LQG methods and quantization of the SUSY constraint
• On the gauge reduction with respect to simplicity constraint in all dimensional loop quantum gravity
• Kinematics of arbitrary spin matter fields in loop quantum gravity
• On the Hamiltonian formalism of the Tetrad-Gravity with fermions
• Hamiltonian Theory: generalizations to higher dimensions, supersymmetry and modified gravity