Abstract
It is well known that linearized gravity in spacetimes with compact Cauchy surfaces and continuous symmetries suffers from linearization instabilities: solutions to classical linearized gravity in such a spacetime must satisfy so-called linearization stability conditions (or constraints) for them to extend to solutions in the full non-linear theory. Moncrief investigated implications of these conditions in linearized quantum gravity in such background spacetimes and found that the quantum linearization stability constraints lead to the requirement that all physical states must be invariant under the symmetries generated by these constraints. He studied these constraints for linearized quantum gravity in flat spacetime with the spatial sections of toroidal topology in detail. Subsequently, his result was reproduced by the method of group-averaging. In this paper the quantum linearization stability conditions are studied for simple supergravity in this spacetime. In addition to the linearization stability conditions corresponding to the spacetime symmetries, i.e. spacetime translations, there are also fermionic linearization stability conditions corresponding to the background supersymmetry. We construct all states satisfying these quantum linearization stability conditions, including the fermionic ones, and show that they are obtained by group-averaging over the supergroup of the global supersymmetry of this theory.
Highlights
In physics, the equations of interest are frequently non-linear and difficult to solve
In addition to the linearization stability conditions corresponding to the spacetime symmetries, i.e. spacetime translations, there are fermionic linearization stability conditions corresponding to the background supersymmetry
We have shown that all states satisfying both the bosonic and fermionic quantum linearization stability conditions (QLSCs) can be obtained in this manner
Summary
The equations of interest are frequently non-linear and difficult to solve. If one considers four-dimensional N = 1 simple supergravity [26, 27] on this background spacetime, it is not difficult to see that there are additional fermionic LSCs. The purpose of this paper is to find all states satisfying the bosonic and fermionic QLSCs and show that a Hilbert space of these states can be constructed using the group-averaging procedure over the supergroup of symmetries of the linearized theory. In supergravity there is a spinor supercharge Qα, α = 1, 2, 3, 4, associated with a global supersymmetry variation, and it is known that this supercharge can be written as an integral over two-dimensional surface at infinity of a Cauchy surface in asymptotically-flat spacetime [28] This fact again implies that in static three-torus space there are quadratic constraints on the linearized theory corresponding to the vanishing of the supercharge, which are the fermionic LSCs. In this paper we study these fermionic LSCs together with the bosonic ones in linearized quantum supergravity. We follow the conventions of reference [29] throughout this paper
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