1 Introduction

Since its inception, it has been clear that General Relativity has many striking similarities to gauge theories. Both are based on the idea of local symmetry and therefore share a number of formal properties. Nevertheless, their dynamical behavior can be quite different. While Maxwell electrodynamics describes a long-range force similar to the situation with gravity, the non-Abelian gauge theories used to describe the weak and strong nuclear forces have rather different behaviors. Quantum chromodynamics, which describes the strong nuclear forces, for example, exhibits confinement of particles carrying the non-Abelian gauge charges. Certainly, there is no obvious corresponding property for gravity. Moreover, consistent quantum gauge theories have existed for a half century, but as yet no satisfactory quantum field theory of gravity has been constructed; indeed, there are good arguments suggesting that it is not possible to do so. The structures of the Lagrangians are also rather different: The non-Abelian Yang–Mills Lagrangian contains only up to four-point interactions while the Einstein–Hilbert Lagrangian contains infinitely many.

Despite these differences, string theory teaches us that gravity and gauge theories can, in fact, be unified. The Maldacena conjecture [95, 2], for example, relates the weak coupling limit of a gravity theory on an anti-de Sitter background to a strong coupling limit of a special supersymmetric gauge field theory. There is also a long history of papers noting that gravity can be expressed as a gauging of Lorentz symmetry [135, 83, 79], as well as examples of non-trivial similarities between classical solutions of gravity and non-Abelian gauge theories [126]. In this review a different, but very general, relationship between the weak coupling limits of both gravity and gauge theories will be described. This relationship allows gauge theories to be used directly as an aid for computations in perturbative quantum gravity.

The relationship discussed here may be understood most easily from string perturbation theory. At the semi-classical or “tree-level”, Kawai, Lewellen, and Tye (KLT) [86] derived a precise set of formulas expressing closed string amplitudes in terms of sums of products of open string amplitudes. In the low-energy limit (i.e. anywhere well below the string scale of 10¹⁹ GeV) where string theory effectively reduces to field theory, the KLT relations necessarily imply that similar relations must exist between amplitudes in gravity and gauge field theories: At tree-level in field theory, graviton scattering must be expressible as a sum of products of well defined pieces of non-Abelian gauge theory scattering amplitudes. Moreover, using string based rules, four-graviton amplitudes with one quantum loop in Einstein gravity were obtained in a form in which the integrands appearing in the expressions were given as products of integrands appearing in gauge theory [25, 55]. These results may be interpreted heuristically as

The KLT relations hold at the semi-classical level, i.e. with no quantum loops. In order to exploit the KLT relations in quantum gravity, one needs to completely reformulate the quantization process; the standard methods starting either from a Hamiltonian or a Lagrangian provide no obvious means of exploiting the KLT relations. There is, however, an alternative approach based on obtaining the quantum loop contributions directly from the semi-classical tree-level amplitudes by using -dimensional unitarity [16, 17, 28, 20, 115]. These same methods have also been applied to non-trivial calculations in quantum chromodynamics (see e.g. Refs. [28, 21, 12]) and in supersymmetric gauge theories (see e.g. Refs. [16, 17, 29, 19]). In a sense, they provide a means for obtaining collections of quantum loop-level Feynman diagrams without direct reference to the underlying Lagrangian or Hamiltonian. The only inputs with this method are the -dimensional tree-level scattering amplitudes. This makes the unitarity method ideally suited for exploiting the KLT relations.

An interesting application of this method of perturbatively quantizing gravity is as a tool for investigating the ultra-violet behavior of gravity field theories. Ultraviolet properties are one of the central issues of perturbative quantum gravity. The conventional wisdom that quantum field theories of gravity cannot possibly be fundamental rests on the apparent non-renormalizability of these theories. Simple power counting arguments strongly suggest that Einstein gravity is not renormalizable and therefore can be viewed only as a low energy effective field theory. Indeed, explicit calculations have established that non-supersymmetric theories of gravity with matter generically diverge at one loop [132, 43, 42], and pure gravity diverges at two loops [66, 136]. Supersymmetric theories are better behaved with the first potential divergence occurring at three loops [39, 82, 81]. However, no explicit calculations have as yet been performed to directly verify the existence of the three-loop supergravity divergences.

The method described here for quantizing gravity is well suited for addressing the issue of the ultraviolet properties of gravity because it relates overwhelmingly complicated calculations in quantum gravity to much simpler (though still complicated) ones in gauge theories. The first application was for the case of maximally supersymmetric gravity, which is expected to have the best ultra-violet properties of any theory of gravity. This analysis led to the surprising result that maximally supersymmetric gravity is less divergent [19] than previously believed based on power counting arguments [39, 82, 81]. This lessening of the power counting degree of divergence may be interpreted as an additional symmetry unaccounted for in the original analysis [129]. (The results are inconsistent, however, with an earlier suggestion [74] based on the speculated existence of an unconstrained covariant off-shell superspace for supergravity, which in implies finiteness up to seven loops. The non-existence of such a superspace was already noted a while ago [81].) The method also led to the explicit construction of the two-loop divergence in eleven-dimensional supergravity [19, 40, 41, 15]. More recently, it aided the study of divergences in type I supergravity theories [54] where it was noted that they factorize into products of gauge theory factors.

Other applications include the construction of infinite sequences of amplitudes in gravity theories. Given the complexity of gravity perturbation theory, it is rather surprising that one can obtain compact expressions for an arbitrary number of external legs, even for restricted helicity or spin configurations of the particles. The key for this construction is to make use of previously known sequences in quantum chromodynamics. At tree-level, infinite sequences of maximally helicity violating amplitudes have been obtained by directly using the KLT relations [10, 14] and analogous quantum chromodynamics sequences. At one loop, by combining the KLT relations with the unitarity method, additional infinite sequences of gravity and super-gravity amplitudes have also been obtained [22, 23]. They are completely analogous to and rely on the previously obtained infinite sequences of one-loop gauge theory amplitudes [11, 16, 17]. These amplitudes turn out to be also intimately connected to those of self-dual Yang–Mills [143, 53, 94, 93, 4, 30, 33] and gravity [108, 52, 109]. The method has also been used to explicitly compute two-loop supergravity amplitudes [19] in dimension , that were then used to check M-theory dualities [68].

Although the KLT relations have been exploited to obtain non-trivial results in quantum gravity theories, a derivation of these relations from the Einstein–Hilbert Lagrangian is lacking. There has, however, been some progress in this regard. It turns out that with an appropriate choice of field variables one can separate the space-time indices appearing in the Lagrangian into ‘left’ and ‘right’ classes [124, 123, 125, 26], mimicking the similar separation that occurs in string theory. Moreover, with further field redefinitions and a non-linear gauge choice, it is possible to arrange the off-shell three-graviton vertex so that it is expressible in terms of a sum of squares of Yang–Mills three-gluon vertices [26]. It might be possible to extend this more generally starting from the formalism of Siegel [124, 123, 125], which contains a complete gravity Lagrangian with the required factorization of space-time indices.

This review is organized as follows. In Section 2 the Feynman diagram approach to perturbative quantum gravity is outlined. The Kawai, Lewellen, and Tye relations between open and closed string tree amplitudes and their field theory limit are described in Section 3. Applications to understanding and constructing tree-level gravity amplitudes are also described in this section. In Section 4 the implications for the Einstein–Hilbert Lagrangian are presented. The procedure for obtaining quantum loop amplitudes from gravity tree amplitudes is then given in Section 5. The application of this method to obtain quantum gravity loop amplitudes is described in Section 6. In Section 7 the quantum divergence properties of maximally supersymmetric supergravity obtained from this method are described. The conclusions are found in Section 8.

There are a number of excellent sources for various subtopics described in this review. For a recent review of the status of quantum gravity the reader may consult the article by Carlip [31]. The conventional Feynman diagram approach to quantum gravity can be found in the Les Houches lectures of Veltman [138]. A review article containing an early version of the method described here of using unitarity to construct complete loop amplitudes is Ref. [20]. Excellent reviews containing the quantum chromodynamics amplitudes used to obtain corresponding gravity amplitudes are the ones by Mangano and Parke [99], and by Lance Dixon [48]. These reviews also provide a good description of helicity techniques which are extremely useful for explicitly constructing scattering amplitude in gravity and gauge theories. Broader textbooks describing quantum chromodynamics are Refs. [107, 141, 58]. Chapter 7 of Superstring Theory by Green, Schwarz, and Witten [70] contains an illuminating discussion of the relationship of closed and open string tree amplitudes, especially at the four-point level. A somewhat more modern description of string theory may be found in the book by Polchinski [110, 111]. Applications of string methods to quantum field theory are described in a recent review by Schubert [120].