Puzzles in 3D Off-Shell Geometries via VTQFT

TL;DR

The work investigates how Virasoro TQFT handles off-shell 3D geometries, focusing on Seifert manifolds and multi-boundary torus-wormholes. By comparing VTQFT extrapolations to the Maxfield-Turiaci program and to Cotler-Jensen JT gravity results, the paper identifies the mapping class group's omission as the core source of discrepancy. Through dimensional reduction to JT gravity and two VTQFT computations for the 3-boundary wormhole, it shows that on-shell results are robust while off-shell extrapolations are sensitive to global diffeomorphism data. The results suggest that including mapping class-group data is essential for a correct 3D off-shell VTQFT program and point toward refined extrapolation schemes or alternative 2D perspectives to capture spectral statistics beyond OPE data. This has implications for understanding negative-density-of-states issues and the broader holographic interpretation of 3D gravity ensembles.

Abstract

We point out a difficulty with a naive application of Virasoro TQFT methods to compute path integrals for two types of off-shell 3-dimensional geometries. Maxfield-Turiaci proposed solving the negativity problem of pure 3d gravity by summing over off-shell geometries known as Seifert manifolds. We attempt to compute Seifert manifolds using Virasoro TQFT. Our results don't match completely with Maxfield-Turiaci. We trace the discrepancies to not including the mapping class group properly. We also compute a 3-boundary torus-wormhole by extrapolating from an on-shell geometry. We encounter challenges similar to those observed in the comparison between the genuine off-shell computation of a torus-wormhole by Cotler-Jensen and the extrapolation from an on-shell configuration.