One-loop exactness of the cusp partition function in 3d gravity

Establish whether higher-loop quantum corrections to the gravitational partition function on the doubly infinite cusp geometry H^3/(Z × Z) vanish or can be absorbed into a rescaling of the cusp fugacity μ, thereby proving that the cusp partition function is one-loop exact.

Background

The paper addresses divergences in the AdS3 gravity path integral arising from accumulation points in the spectrum of hyperbolic 3-manifold volumes. The authors propose to renormalize these divergences by including a contribution from the limiting cusped manifold and tuning a fugacity per cusp to cancel the divergence.

A key step in their proposal is the computation of the cusp’s one-loop determinant, which after a modular-invariant regularization yields a contribution proportional to (√τ2 |η(τ)|2)^{-1}. Using this, they show the cusp can serve as a counterterm matching the divergence found in the modular Poincaré sum over SL(2,ℤ) black holes.

However, the renormalization argument relies on the assumption that the cusp partition function is one-loop exact. The authors explicitly note that higher-loop effects must be checked to validate the exactness or to determine whether they can be absorbed into μ, making this a central unresolved question for the consistency of the proposal.