AI 中文总结
该研究探讨AdS₃/CFT₂的系综全息是否可通过平均Virasoro TQFT的拓扑边界条件实现，证明一维亏格下Virasoro TQFT无类似纳雷恩的系综机制，需额外信息构建非平凡系综解释。
AI 中文摘要
受三维引力与加倍Virasoro拓扑量子场论（TQFT）之间提出的关联以及相关的系综全息计划的启发，我们探究AdS₃/CFT₂中的系综全息是否可理解为对加倍Virasoro TQFT拓扑边界条件变化得到的绝对二维共形场论（CFT）的平均。我们证明，在一维亏格且普通非简并扇区中，每个在辅助L²标记空间上有界作用的真空归一化模不变一维亏格配对均为对角型。在普通对角化真空-连续谱假设下，即使是初始奇异或无界的、满足两个真空边际条件的逐元素正模不变博雷尔测度配对，也会得出相同结论。因此，在这些解析类中，Virasoro TQFT无法基于普通拓扑边界的不同非平凡环面配对平均来实现类似纳雷恩的系综机制；任何非平凡系综解释都必须涉及一维亏格不可见的信息、额外扇区或真正广义边界条件。技术上，该证明将Virasoro S变换与T变换等同于亚纯群的偶魏尔表示，并应用了Cowling-Steger格限制定理；我们还给出了若干显式候选边界条件类的初等证明，并对该一般结果进行了数值检验。
英文摘要
Motivated by the proposed relation between 3D gravity and the doubled Virasoro TQFT, and by the associated program of ensemble holography, we ask whether ensemble holography in AdS$_3$/CFT$_2$ can be understood as an average over absolute 2D CFTs obtained by varying the topological boundary condition of the doubled Virasoro TQFT. We show that, at genus one and in the ordinary nondegenerate sector, every vacuum-normalized, modular-invariant genus-one pairing that acts boundedly on the auxiliary $L^2$ label space is diagonal. Under the ordinary block-diagonal vacuum--continuum ansatz, the same conclusion holds for entrywise-positive, modular-invariant Borel-measure pairings satisfying the two vacuum marginals, without assuming absolute continuity or boundedness. The bounded class and this positive-measure class therefore contain no distinct nontrivial torus pairings to average. Any nontrivial Virasoro boundary ensemble must use genus-one pairings outside these classes, involve additional sectors or vacuum--continuum couplings, or distinguish its members through information not captured by the genus-one pairing. Technically, the proof identifies the Virasoro $S$ and $T$ transformations with the even Weil representation of the metaplectic group and applies the Cowling--Steger lattice-restriction theorem. We also give elementary proofs for several explicit classes of candidate boundary conditions and perform numerical checks of the general result.
Comments48 pages, 3 figures; ancillary code included. v2: revised title and physical interpretation to correct an overclaim in v1; added references
