Ultraviolet Behavior of N=8 Supergravity

TL;DR

This work investigates the perturbative ultraviolet behavior of ${\cal N}=8$ supergravity using on-shell techniques that connect gravity to gauge theory. By employing the unitarity method and KLT relations, the author shows that up to four loops the four-point amplitude exhibits cancellations that render it no worse than the corresponding ${\cal N}=4$ SYM case, with explicit three- and four-loop results indicating finiteness in dimensions $D<\tfrac{11}{2}$ and a three-loop divergence in $D=6$ associated with a ${\cal D}^6R^4$ counterterm. The analyses rely on the double-copy structure (gravity as a square of gauge theory) and generalized unitarity with maximal cuts, and they reveal remarkable cancellations beyond naive power counting. While five-loop behavior remains unsettled, these findings provide strong evidence that ${\cal N}=8$ supergravity may be perturbatively finite as a quantum theory of gravity in four dimensions, offering valuable insights for the quest toward consistent quantum gravity models.

Abstract

In these lectures I describe the remarkable ultraviolet behavior of N=8 supergravity, which through four loops is no worse than that of N=4 super-Yang-Mills theory (a finite theory). I also explain the computational tools that allow multi-loop amplitudes to be evaluated in this theory - the KLT relations and the unitarity method - and sketch how ultraviolet divergences are extracted from the amplitudes.

Figures (12)

• Figure 1: Schematic depiction of the KLT relations. The closed-string world-sheet, a sphere, can be thought of as two copies of the open-string world-sheet, a disk. Vertex operator insertions are marked with $\times$'s.
• Figure 2: Unitarity relations for the four-point amplitude at one and two loops. The number of holes in a blob indicates the number of loops in the corresponding amplitude.
• Figure 3: Generalized unitarity at one loop: (a) the ordinary two-particle cut imposes two constraints on the loop-momentum; (b) the triple cut imposes three constraints and is sensitive to triangle coefficients; (c) the quadruple cut imposes four constraints (freezing all four components of $\ell_1^\mu$) and is sensitive to box coefficients.
• Figure 4: An example of multi-loop generalized unitarity. The one-loop five-point amplitude, appearing on the right side of the ordinary cut, is further cut into products of trees, in three inequivalent ways.
• Figure 5: Example of a real-momentum generalized cut generating several maximal cuts; the latter contain only three-point tree amplitudes.