Approaches to Quantum Gravity

Gauge/gravity duality 173decomposed into spherical harmonics, which can be described as symmetric tracelesstensors on R 6 : T i··· j X i ···X j . Restricted to the unit sphere one gets a basis offunctions. Recall that the gauge theory has six scalars and the SO(6) symmetry ofrotating the ϕ i . So the operators T i··· j ϕ i ···ϕ j give information about position onS 5 . Four of the remaining directions are explicitly present in the gauge theory, andthe radial direction corresponds to the energy scale in the gauge theory.In the gauge theory the expectation values of local operators (gauge invariantproducts of the N = 4 fields and their covariant derivatives) provide one naturalset of observables. It is convenient to work with the generating functional for theseexpectation values by shifting the Lagrangian densityL(x) → L(x) + ∑ I J I (x)O I (x), (10.3)where O I is some basis of local operators and J I (x) are arbitrary functions. Sincewe are taking products of operators at a point, we are perturbing the theory in theultraviolet, which according to the energy–radius relation maps to the AdS boundary.Thus the duality dictionary relates the gauge theory generating functional to agravitational theory in which the boundary conditions at infinity are perturbed in aspecified way [16; 42]. As a further check on the duality, all three-point interactionswere shown to agree [28].The linearized supergravity excitations map to gauge invariant states of thegauge bosons, scalars, and fermions, but in fact only to a small subset of these;in particular, all the supergravity states live in special small multiplets of the superconformalsymmetry algebra. Thus the dual to the gauge theory contains muchmore than supergravity. The identity of the additional degrees of freedom becomesparticularly clear if one looks at highly boosted states, those having large angularmomentum on S 5 and/or AdS 5 [5; 17]. The fields of the gauge theory thenorganize naturally into one-dimensional structures, coming from the Yang–Millslarge-N c trace: they correspond to the excited states of strings. In some cases, onecan even construct a two dimensional sigma model directly from the gauge theoryand show that it agrees (at large boost) with the sigma model describing stringsmoving in AdS 5 × S 5 [27].Thus, by trying to make sense of the assertion at the beginning, we are forced to“discover” string theory. We can now state the duality in its full form [30].Four-dimensional N = 4 supersymmetric SU(N c ) gauge theory is equivalent to IIB stringtheory with AdS 5 × S 5 boundary conditions.The need for strings (though not the presence of gravity!) was already anticipatedby ’t Hooft [20], based on the planar structure of the large-N c Yang–Mills perturbationtheory; the AdS/CFT duality puts this into a precise form. It also fits