Physical States in d=3,N=2 Supergravity

TL;DR

This paper investigates the quantum states of three-dimensional N=2 supergravity, revealing that wave functionals depend on complex moduli spaces and are not expandable in fermionic coordinates, offering insights relevant to higher-dimensional theories.

Contribution

It provides a detailed analysis of the wave functionals in 3D N=2 supergravity, clarifying their structure and implications for quantum constraints in topological supergravity.

Findings

Wave functionals depend on non-Hausdorff moduli spaces.

Wave functionals cannot be expanded in fermionic coordinates.

Solutions can serve as lowest-order approximations in quantum constraints.

Abstract

To clarify some issues raised by D'Eath's recent proposal for the physical states of $N=1$ supergravity in four dimensions, we study pure (topological) $N=2$ supergravity in three dimensions, which is formally very similar, but much easier to solve. The wave functionals solving the quantum constraints can be understood in terms of arbitrary functions on the space of moduli and supermoduli, which is not Hausdorff. We discuss the implications for the wave functionals and show that these are not amenable to expansions in fermionic coordinates, but can serve as lowest-order solutions to the quantum constraints in an expansion in $\hbar$ in more realistic theories.