Progress on Ultraviolet Finiteness of Supergravity

TL;DR

The paper surveys progress toward perturbative ultraviolet finiteness of N=8 supergravity in four dimensions using on-shell and unitarity methods. It highlights gravity–gauge theory relations (KLT/twist), no-triangle cancellations at one loop, and a three-loop four-point amplitude whose UV behavior mirrors N=4 SYM, suggesting possible all-loop cancellations beyond supersymmetry. These findings imply a potential mechanism driving UV finiteness in gravity, with implications for the gravity–gauge unification and guidance for future high-loop calculations. Continued work, including four-loop analyses, is essential to confirm finiteness and uncover underlying symmetry or dynamical principles.

Abstract

In this lecture we summarize recent calculations pointing to the possible ultraviolet finiteness of N = 8 supergravity in four dimensions. We outline the modern unitarity method, which enables multiloop calculations in this theory and allows us to exploit a remarkable relation between tree-level gravity and gauge-theory amplitudes. We also describe a link between observed cancellations at loop level and improved behavior of tree-level amplitudes under large complex deformations of momenta.

Figures (8)

• Figure 1: Gauge theories have three- and four-point vertices in a Feynman diagrammatic description.
• Figure 2: Gravity theories have an infinite number of higher-point contact interactions in a Feynman diagrammatic description.
• Figure 3: A sample diagram at three loops. A three-graviton vertex is quadratic in the momenta while a gauge theory vertex in linear in the momenta, suggesting that gravity theories are worse behaved in the ultraviolet than gauge theories.
• Figure 4: Examples of generalized cuts for determining a three-loop four-point amplitude. Each blob represents an on-shell tree amplitude, and the displayed intermediate lines are all on shell.
• Figure 5: Any one-loop $n$-point amplitude can be expressed as a linear combination of scalar box, triangle and bubble integrals. In ${{\cal N}=4}$ super-Yang-Mills and ${{\cal N}=8}$ supergravity, after reducing all integrals to this basis, the coefficients of all triangle and bubble integrals vanish.