Progress on Ultraviolet Finiteness of Supergravity

Abstract

AbstractIn this lecture we summarize recent calculations pointing to the possible ultra-violet finiteness of N = 8 supergravity in four dimensions. We outline the modernunitarity method, which enables multiloop calculations in this theory and allows usto exploit a remarkable relation between tree-level gravity and gauge-theory am-plitudes. We also describe a link between observed cancellations at loop level andimproved behavior of tree-level amplitudes under large complex deformations ofmomenta. 1 Introduction For over 25 years the prevailing wisdom has been that it is impossible to construct a per-turbatively ultraviolet finite point-like quantum field theory of gravity in four dimensions(see e.g. refs. [1]). In this lecture we describe recent concrete calculations that call intoquestion this belief.Of all unitary quantum gravity field theories, maximally supersymmetric N = 8 su-pergravity [2] is the most promising one to investigate for possible ultraviolet finiteness.Its high degree of supersymmetry suggests that it has the best ultraviolet properties ofany gravity field theory with two derivative couplings. Moreover, with the modern uni-tarity method [3, 4, 5, 6, 7, 8, 9], the high degree of supersymmetry can be exploited togreatly simplify calculations. In fact, the striking simplicity of the theory led to the recentsuggestion that N = 8 supergravity may in a sense be the simplest quantum field the-ory [10]. The unitarity method allows us to exploit a remarkable relation between gravityand gauge-theory tree amplitudes [11, 12, 13], allowing us to map gravity calculationsinto algebraically simpler gauge-theory calculations.In a classic paper ’t Hooft and Veltman showed that gravity coupled to matter gener-ically diverges at one loop in four dimensions [14, 15]. Due to the dimensionful natureof the coupling, the divergences cannot be absorbed by a redefinition of the originalparameters of the Lagrangian, rendering the theory non-renormalizable. Pure Einsteingravity does not possess a viable counterterm at one loop, delaying the divergence to twoloops [14, 16, 17]. The two-loop divergence of pure Einstein gravity was established byGoroff and Sagnotti and by van de Ven, through direct computation [18, 19].