We generalize the recently discovered relationship between JT gravity and double-scaled random matrix theory to the case that the boundary theory may have time-reversal symmetry and may have fermions with or without supersymmetry. The matching between variants of JT gravity and matrix ensembles depends on the assumed symmetries. Time-reversal symmetry in the boundary theory means that unorientable spacetimes must be considered in the bulk. In such a case, the partition function of JT gravity is still related to the volume of the moduli space of conformal structures, but this volume has a quantum correction and has to be computed using Reidemeister–Ray–Singer “torsion.” Presence of fermions in the boundary theory (and thus a symmetry $(-1)^\mathsf{F}$) means that the bulk has a spin or pin structure. Supersymmetry in the boundary means that the bulk theory is associated to JT supergravity and is related to the volume of the moduli space of super Riemann surfaces rather than of ordinary Riemann surfaces. In all cases we match JT gravity or supergravity with an appropriate random matrix ensemble. All ten standard random matrix ensembles make an appearance—the three Dyson ensembles and the seven Altland–Zirnbauer ensembles. To facilitate the analysis, we extend to the other ensembles techniques that are most familiar in the case of the original Wigner–Dyson ensemble of hermitian matrices. We also generalize Mirzakhani’s recursion for the volumes of ordinary moduli space to the case of super Riemann surfaces.

中文翻译：

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JT 引力和随机矩阵理论的集合

我们将最近发现的 JT 引力与双标度随机矩阵理论之间的关系推广到边界理论可能具有时间反转对称性并且可能具有具有或不具有超对称性的费米子的情况。JT 引力变体与矩阵系综之间的匹配取决于假设的对称性。边界理论中的时间反转对称性意味着必须在整体上考虑不可定向的时空。在这种情况下，JT 引力的配分函数仍然与共形结构的模空间体积有关，但该体积具有量子校正，必须使用 Reidemeister-Ray-Singer“扭转”计算。边界理论中费米子的存在（因此对称性 $(-1)^\mathsf{F}$）意味着体块具有自旋或针结构。边界中的超对称性意味着体积理论与 JT 超引力有关，并且与超黎曼曲面的模空间的体积有关，而不是与普通黎曼曲面的模空间体积有关。在所有情况下，我们将 JT 重力或超重力与适当的随机矩阵系综相匹配。所有十个标准随机矩阵系综都出现了——三个戴森系综和七个 Altland-Zirnbauer 系综。为了便于分析，我们扩展到在厄密矩阵的原始 Wigner-Dyson 系综的情况下最熟悉的其他系综技术。我们还将 Mirzakhani 对普通模空间体积的递归推广到超黎曼曲面的情况。