Abstract
In this paper, we study a class of symmetry reduced models of mathcal{N} = 1 super- gravity using self-dual variables. It is based on a particular Ansatz for the gravitino field as proposed by D’Eath et al. We show that the essential part of the constraint algebra in the classical theory closes. In particular, the (graded) Poisson bracket between the left and right supersymmetry constraint reproduces the Hamiltonian constraint.For the quantum theory, we apply techniques from the manifestly supersymmetric approach to loop quantum supergravity, which yields a graded analog of the holonomy-flux algebra and a natural state space.We implement the remaining constraints in the quantum theory. For a certain subclass of these models, we show explicitly that the (graded) commutator of the supersymmetry constraints exactly reproduces the classical Poisson relations. In particular, the trace of the commutator of left and right supersymmetry constraints reproduces the Hamilton constraint operator. Finally, we consider the dynamics of the theory and compare it to a quantization using standard variables and standard minisuperspace techniques.
Highlights
For the quantum theory, we apply techniques from the manifestly supersymmetric approach to loop quantum supergravity, which yields a graded analog of the holonomy-flux algebra and a natural state space
It is based on a particular Ansatz for the gravitino field as proposed by D’Eath et al We show that the essential part of the constraint algebra in the classical theory closes
Equation (3.76) provides a very strong relation between the Hamiltonian and supersymmetry constraint which will play a central role in section 4.4 in the construction of the physical sector of the kinematical Hilbert space and the study of the resulting dynamics of the theory
Summary
With ΣAB and ΣA B the self-dual anti self-dual part of ΣAA BB , respectively, given by
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