The Ashtekar complex canonical transformation for supergravity

We consider the specialization to spatially homogenous solutions of the Jacobson formulation of N=1 canonical supergravity in terms of Ashtekar's new variables. We find that the classical Poisson algebra of the supersymmetry constraints is preserved by this specialization only for Bianchi type A models. The quantization of supersymmetric Bianchi type A models is carried out in the triad representation. We find the physical states of this quantum theory. Since we are missing a suitable inner product on these physical states, our results are only formal.

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“…where we denote with h the determinant of h ab , and the chain rule, T r(σ a δ/δσ b ) = 2h ac δ/δh bc , equation (28) can be put in the form,…”

Section: Bosonic Statesmentioning

confidence: 99%

“…The quantity F is the bosonic part of the homogenous specialization of the generating functional of the canonical transformation from the triad extended ADM phase space to Jacobson's [28] .…”

Section: Bosonic Statesmentioning

confidence: 99%

Super-minisuperspace and new variables

Capovilla

Guven

1994

Class. Quantum Grav.

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“…42) which can be verified to be unitary and strongly continuous in Λ. It maybe verified explicitly ] is self-adjoint by Stone's theorem…”

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confidence: 90%

Towards loop quantum supergravity (LQSG): I. Rarita–Schwinger sector

Bodendorfer

Thiemann

Thurn

2013

Class. Quantum Grav.

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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

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“…The original Ashtekar's self-dual canonical grav ity permits also a Lagrangian formulation [3] - [4]. The supersymmetric extension of this Lagrangia formulation, which is equivalent to the simple real supergravity, was proposed by Jacobson [5], an the corresponding Ashtekar complex canonical transform was given by Gorobey et al [6].…”

Section: Introductionmentioning

confidence: 99%

Superalgebra and conservative quantities in N = 1 complex supergravity

Feng

Wang

2000

Class. Quantum Grav.

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The N = 1 self-dual supergravity has SL(2, C) symmetry and the lefthanded and right-handed local supersymmetries. These symmetries result in SU(2) charges as the angular-momentum and the supercharges. The model possesses also the invariance under the general translation transforms and this invariance leads to the energy-momentum. All the definitions are generally covariant. As the SU(2) charges and the energymomentum we obtained previously constituting the 3-Poincare algebra in the Ashtekar'complex gravity, the SU(2) charges, the supercharges and the energy-momentum in simple supergravity also restore the super-Poincare algebra, and this serves to support the reasonableness of their interpretations. PACS number(s): 11.30.Cp,11.30.Pb,04.20.Me,04.65.+e. superalgebra complex supergravity 1 Introduction The study of self-dual gravities has drawn much attention in the past decade since the discover of Ashtekar's new variables, in terms of which the constraints can be greatly simplified[1]-[2]. Th new phase variables consist of densitized SU(2) soldering formsẽ i A B from which a metric density obtained according to the definition q ij = −Trẽ iẽj , and a complexified connection A iA B which carrie the momentum dependence in its imaginary part. The original Ashtekar's self-dual canonical grav ity permits also a Lagrangian formulation[3] -[4]. The supersymmetric extension of this Lagrangia formulation, which is equivalent to the simple real supergravity, was proposed by Jacobson[5], an the corresponding Ashtekar complex canonical transform was given by Gorobey et al[6]. In our previous works, we have obtained the SU(2) charges and the energy-momentum in th Ashtekar's formulation of Einstein gravity[7]-[8] and they are closely related to the angular-momentu [11] and the energy-momentum [12] in the vierbein formalism of Einstein gravity. The fact that th algebra formed by their Poisson brackets do constitute the 3-Poincare algebra on the Cauchy surfac supports from another aspect that their definitions are reasonable.Out of the same reason, the definitions of SU(2) charges, which are to be interpreted as th angular-momentum, the supercharges and the energy-momentum are also interesting and importan aspects in the simple self-dual supergravity. In this paper, we will exploit the SL(2, C) invarianc the left-handed and right-handed supersymmetry and the invarinace under the general translatio transform[12] to obtain the conservative charges under consideration. This paper is arranged a follows. In section 2, we will give a brief review of the N = 1 self-dual supergravity. In sectio 3, we will derive the SU(2) charges from the original Lagrangian of Jacobson. In section 4, w derive the energy-momentum from a slightly different Lagrangian and the general translation. I section 5, we derive the supercharges from the invariance under left-handed and right-handed loc supersymmetric transforms. The last section is devoted to summary and discussions.

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The Ashtekar complex canonical transformation for supergravity

Abstract: The complex canonical transformation introduced by Ashtekar (1987) in general relativity is extended to simple supergravity.

Cited by 9 publications

References 7 publications

Super-minisuperspace and new variables

Super-minisuperspace and new variables

Towards loop quantum supergravity (LQSG): I. Rarita–Schwinger sector

Superalgebra and conservative quantities in N = 1 complex supergravity

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