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 $\mathcal{N}=1$ 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.

中文翻译：

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三环面上的超引力：超群的量子线性化不稳定性

众所周知，具有紧凑柯西表面和连续对称性的时空中的线性化重力存在线性化不稳定性：在这样的时空中经典线性化重力的解必须满足所谓的线性化稳定性条件（或约束）才能将它们扩展到完全非线性理论。Moncrief 研究了这些条件在这种背景时空中的线性化量子引力中的含义，并发现量子线性化稳定性约束导致所有物理状态必须在这些约束产生的对称性下保持不变的要求。他详细研究了平面时空中线性化量子引力的这些约束，并详细研究了环形拓扑的空间截面。随后，他的结果通过组平均的方法再现。本文研究了该时空中$\mathcal{N}=1$简单超引力的量子线性化稳定性条件。除了时空对称性即时空平移对应的线性化稳定性条件外，还有背景超对称性对应的费米子线性化稳定性条件。我们构建了满足这些量子线性化稳定性条件的所有状态，包括费米子状态，并表明它们是通过对该理论全局超对称超群的群平均获得的。还有对应于背景超对称性的费米子线性化稳定性条件。我们构建了满足这些量子线性化稳定性条件的所有状态，包括费米子状态，并表明它们是通过对该理论全局超对称超群的群平均获得的。还有对应于背景超对称性的费米子线性化稳定性条件。我们构建了满足这些量子线性化稳定性条件的所有状态，包括费米子状态，并表明它们是通过对该理论全局超对称超群的群平均获得的。