Genus-One Rigidity of Bounded Virasoro Pairings for 3D Gravity

Abstract

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.