Matrix Models , Emergent Gravity , and Gauge Theory

Matrix models of Yang-Mills type induce an effective gravity theory on 4-dimensional branes, which are considered as models for dynamical space-time. We review recent progress in the understanding of this emergent gravity. The metric is not fundamental but arises effectively in the semi-classical limit, along with nonabelian gauge fields. This leads to a mechanism for protecting certain geometries from corrections due to the vacuum energy. 1 Background and motivation Quantum field theory and general relativity (GR) provide the basis of our present understanding of fundamental matter and interactions. In spite of the success of these two theories, there is up to now no satisfactory way to reconcile them in a consistent quantum theory. In particular, quantum mechanics combined with GR strongly suggests a “foam-like” or quantum structure at the Planck scale 10−33 cm, where quantum fluctuations of space-time are expected to be important. While some kind of quantum structure of space-time indeed arises e.g. in string theory, a satisfactory understanding is still missing. The cosmological constant problem should be seen in the same context: the observed tiny (or zero) value of the cosmological constant is in striking contradiction with quantum mechanical expectations, which are off by a factor of order 10120. Reconciling quantum mechanics with gravity is therefore of utmost importance in theoretical physics. In view of these problems, it is natural to consider noncommutative (NC) or quantum spaces as models for space-time. For fixed backgrounds, considerable work has been done in this context, leading to NC field theory [4]. Recently, it was understood that gravity emerges naturally from NC gauge theory, without having to introduce an explicit dynamical metric. Earlier forms of this idea [5, 6] can be cast in concise form for matrix models of Yang-Mills type [8], which describe dynamical quantum spaces. We discuss basic results of this approach. The IKKT model [10] is singled out as a prime candidate for a quantum theory of space-time and matter. 2 The quantization of Poisson manifolds. Space-time in GR is modeled by a 4-dimensional manifold M with metric Gμν(x). The basic assumption of the present approach is that space-time carries an additional Poisson structure {xμ, xν} = θμν(x) (which will be related to the metric in (23)), more precisely that space-time is the quantization Mθ of such a Poisson manifold. In principle, a Poisson structure breaks (local) Lorentz invariance, which may seem incompatible with observation. However, it turns out that θμν does not enter explicitly the effective action of the models discussed here, to leading order in an expansion in θμν . If we assume that the scale of noncommutativity ΛNC defined by det θ μν = ΛNC is at or near the Planck scale, it