Puzzles in 3D Off-Shell Geometries via VTQFT

TL;DR

This paper investigates the application of Virasoro TQFT to off-shell 3D geometries, highlighting discrepancies and challenges in accurately computing path integrals for Seifert manifolds and torus-wormholes.

Contribution

It identifies issues in applying Virasoro TQFT to off-shell geometries and emphasizes the importance of including the mapping class group for correct results.

Findings

Discrepancies due to neglecting the mapping class group.

Challenges in extrapolating from on-shell to off-shell geometries.

Partial mismatch with previous proposals for 3D gravity path integrals.

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.