On the One-Loop Exactness of Gravity Partition Function

TL;DR

This work tests the one-loop exactness of the gravity partition function by extending a perturbative flat-space analysis to arbitrary dimensions and then specializing to D=3. Using a perturbative expansion around flat space, the authors show that two-loop vacuum diagrams vanish in any dimension via dimensional regularization, and that in D=3 the full three-loop contribution cancels after reducing to a single master integral, consistent with a one-loop exact partition function. In contrast, for generic D, many three-loop diagrams survive and contribute to a linear combination of several master integrals, with the total no longer guaranteed to vanish; the coefficients form a dimension-dependent polynomial with a root at D=3, hinting at a unique three-dimensional algebraic mechanism behind the cancellations. The results suggest that one-loop exactness in D=3 has a special origin tied to the dimensionality, warranting further work to uncover the diagrammatic or algebraic structures responsible for these cancellations.

Abstract

In a previous work, we showed that the two- and three-loop contributions to the partition function of three-dimensional gravity in flat space vanish. This is in agreement with the expected one-loop exactness dictated by the underlying symmetry at the quantum level. To highlight the distinctive nature of the $D=3$ case, we extend the three-loop computation to arbitrary spacetime dimensions $D$. In higher dimensions, the number of contributions increases substantially, reinforcing the view that one-loop exactness is a unique feature of three-dimensional gravity.

Figures (8)

• Figure 1: Completely symmetrized expression of the three-graviton vertex.
• Figure 2: Tadpole-like diagram vanishing by dimensional regularization. The same remark apply by replacing the graviton loop by a ghost propagator loop.
• Figure 3: Two-loop diagrams.
• Figure 4: Three-loop contributions
• Figure 5: Snowman-type three-loop diagrams