Conclusions

8 Conclusions

This review described how the notion that gravity $∼$ (gauge theory) $×$ (gauge theory) can be exploited to develop a better understanding of perturbative quantum gravity. The Kawai–Lewellen–Tye (KLT) string theory relations [86 ] give this notion a precise meaning at the semi-classical or tree-level. Quantum loop effects may then be obtained by using $D$ -dimensional unitarity [16 , 17 , 28 , 20 ]. In a sense, this provides an alternative method for quantizing gravity, at least in the context of perturbative expansions around flat space. With this method, gauge theory tree amplitudes are converted into gravity tree amplitudes which are then used to obtain gravity loop amplitudes. The ability to carry this out implies that gravity and gauge theory are much more closely related than one might have deduced by an inspection of the respective Lagrangians.

Some concrete applications were also described, including the computation of the two-loop four-point amplitude in maximally supersymmetric supergravity. The result of this and related computations is that maximal supergravity is less divergent in the ultraviolet than had previously been deduced from superspace power counting arguments [19 , 129 ]. For the case of $D = 4$ , maximal supergravity appears to diverge at five instead of three loops. Another example for which the relation is useful is for understanding the behavior of gravitons as their momenta become either soft or collinear with the momenta of other gravitons. The soft behavior was known long ago [139 ], but the collinear behavior is new. The KLT relations provide a means for expressing the graviton soft and collinear functions directly in terms of the corresponding ones for gluons in quantum chromodynamics. Using the soft and collinear properties of gravitons, infinite sequences of maximally helicity violating gravity amplitudes with a single quantum loop were obtained by bootstrapping [22 , 23 ] from the four-, five-, and six-point amplitudes obtained by direct calculation using the unitarity method together with the KLT relations. Interestingly, for the case of identical helicity, the sequences of amplitudes turn out to be the same as one gets from self-dual gravity [108 , 52 , 109 ].

There are a number of interesting open questions. Using the relationship of gravity to gauge theory one should be able to systematically re-examine the divergence structure of non-maximal theories. Some salient work in this direction may be found in Ref. [54 ], where the divergences of Type I supergravity in $D = 8,10$ were shown to split into products of gauge theory factors. More generally, it should be possible to systematically re-examine finiteness conditions order-by-order in the loop expansion to more thoroughly understand the divergences and associated non-renormalizability of quantum gravity.

An important outstanding problem is the lack of a direct derivation of the KLT relations between gravity and gauge theory tree amplitudes starting from their respective Lagrangians. As yet, there is only a partial understanding in terms of a “left-right” factorization of space-time indices [124 , 123 , 125 , 26 ], which is a necessary condition for the KLT relations to hold. A more complete understanding may lead to a useful reformulation of gravity where properties of gauge theories can be used to systematically understand properties of gravity theories and vice versa. Connected with this is the question of whether the heuristic notion that gravity is a product of gauge theories can be given meaning outside of perturbation theory.

In summary, the perturbative relations between gravity and gauge theory provide a new tool for understanding non-trivial properties of quantum gravity. However, further work will be required to unravel fully the intriguing relationship between the two theories.