Approaches to Quantum Gravity

Gauge/gravity duality 177There is also a field theory interpretation of black hole quasinormal modes [22].A perturbation of the black hole decays with a characteristic time set by the imaginarypart of the lowest quasinormal mode. This should correspond to the timescalefor the gauge theory to return to thermal equilibrium. One can show that the quasinormalmode frequencies are poles in the retarded Green’s function of a certainoperator in the gauge theory. The particular operator depends on the type of fieldused to perturb the black hole [26].Finally, consider the formation and evaporation of a small black hole in aspacetime which is asymptotically AdS 5 × S 5 . By the AdS/CFT correspondence,this process is described by ordinary unitary evolution in the gauge theory. Soblack hole evaporation does not violate quantum mechanics: information is preserved.This also provides an indirect argument against the existence of a “bounce”at the black hole singularity, because the resulting disconnected universe wouldpresumably carry away information.10.3.2 Background independence and emergenceThe AdS/CFT system is entirely embedded in the framework of quantum mechanics.On the gauge theory side we have an explicit Hamiltonian, and states whichwe can think of as gauge invariant functionals of the fields. Thus the gravitationaltheory on the other side is quantum mechanical as well. In particular the metricfluctuates freely except at the AdS boundary. One is not restricted to perturbationsabout a particular background.This is clearly illustrated by a rich set of examples which provide a detailed mapbetween a class of nontrivial asymptotically AdS 5 × S 5 supergravity solutions anda class of states in the gauge theory [29]. These states and geometries both preservehalf of the supersymmetry of AdS 5 ×S 5 itself. On the field theory side, one restrictsto fields that are independent of S 3 and hence reduce to N c × N c matrices. In fact,all the states are created by a single complex matrix, so can be described by aone-matrix model. This theory can be quantized exactly in terms of free fermions,and the states can be labeled by a arbitrary closed curve (the Fermi surface) on aplane. On the gravity side, one considers solutions to ten dimensional supergravityinvolving just the metric and self-dual five form F 5 . The field equations are simplydF 5 = 0andR MN = F MPQRS F N PQRS . (10.9)There exists a large class of stationary solutions to (10.9), which have an SO(4) ×SO(4) symmetry and can be obtained by solving a linear equation. These solutionsare nonsingular, have no event horizons, but can have complicated topology.