Jan Ambjørn

Diseases of triangulated random surface models, and possible cures

Abstract We discuss divergence problems in models of triangulated random surfaces whose action is given, for example, by the surface area. Our results uncover new difficulties in quantizing and regularizing the Nambu-Goto string. We also propose a new class of random surface models with more coercive actions which are expected to have improved behaviour and appear to be accessible to numerical simulations.

Wrapping interactions and a new source of corrections to the spin-chain/string duality

Abstract Assuming that the world-sheet sigma-model in the AdS/CFT correspondence is an integrable quantum field theory, we deduce that there might be new corrections to the spin-chain/string Bethe ansatz paradigm. These come from virtual particles propagating around the circumference of the cylinder and render Bethe ansatz quantization conditions only approximate. We determine the nature of these corrections both at weak and at strong coupling in the near-BMN limit, and find that the first corrections behave qualitatively as wrapping interactions at weak coupling.

The Spectral Dimension of the Universe is Scale Dependent

We measure the spectral dimension of universes emerging from nonperturbative quantum gravity, defined through state sums of causal triangulated geometries. While four-dimensional on large scales, the quantum universe appears two-dimensional at short distances. We conclude that quantum gravity may be “selfrenormalizing” at the Planck scale, by virtue of a mechanism of dynamical dimensional reduction.

Emergence of a 4D world from causal quantum gravity

Causal Dynamical Triangulations in four dimensions provide a background-independent definition of the sum over geometries in nonperturbative quantum gravity, with a positive cosmological constant. We present evidence that a macroscopic four-dimensional world emerges from this theory dynamically.

Reconstructing the universe

We provide detailed evidence for the claim that nonperturbative quantum gravity, defined through state sums of causal triangulated geometries, possesses a large-scale limit in which the dimension of spacetime is four and the dynamics of the volume of the universe behaves semiclassically. This is a first step in reconstructing the universe from a dynamical principle at the Planck scale, and at the same time provides a nontrivial consistency check of the method of causal dynamical triangulations. A closer look at the quantum geometry reveals a number of highly nonclassical aspects, including a dynamical reduction of spacetime to two dimensions on short scales and a fractal structure of slices of constant time.

Non-perturbative Lorentzian Quantum Gravity, Causality and Topology Change

Abstract We formulate a non-perturbative lattice model of two-dimensional Lorentzian quantum gravity by performing the path integral over geometries with a causal structure. The model can be solved exactly at the discretized level. Its continuum limit coincides with the theory obtained by quantizing 2d continuum gravity in proper-time gauge, but it disagrees with 2d gravity defined via matrix models or Liouville theory. By allowing topology change of the compact spatial slices (i.e. baby universe creation), one obtains agreement with the matrix models and Liouville theory.

Matrix model calculations beyond the spherical limit

We propose an improved iterative scheme for calculating higher genus contributions to the multi-loop (or multi-point) correlators and the partition function of the hermitian one matrix model. We present explicit results up to genus two. We develop a version which gives directly the result in the double scaling limit and present explicit results up to genus four. Using the latter version we prove that the hermitian and the complex matrix model are equivalent in the double scaling limit and that in this limit they are both equivalent to the Kontsevich model. We discuss how our results away from the double scaling limit are related to the structure of moduli space.

Multiloop correlators for two-dimensional quantum gravity

Abstract We find explicitly all multi-loop correlators in the complex matrix model to the leading order in 1 N and show that they are identical to the even part of the multi-loop correlators in the hermitean matrix model. The scaling limit for the corresponding macroscopic loop correlators is constructed and agrees with the one of the hermitean model to all orders in 1 N 2 . In particular the double scaling limits of the two models will lead to identical “string equations”.

Three-dimensional simplicial quantum gravity and generalized matrix models

We consider a discrete model of Euclidean quantum gravity in three dimensions based on a summation over random simplicial manifolds. We derive some elementary properties of the model and discuss possible “matrix” models for 3-D gravity.

Finite N matrix models of noncommutative gauge theory

We describe a unitary matrix model which is constructed from discrete analogs of the usual projective modules over the noncommutative torus and use it to construct a lattice version of noncommutative gauge theory. The model is a discretization of the noncommutative gauge theories that arise from toroidal compactification of Matrix theory and it includes a recent proposal for a non-perturbative definition of noncommutative Yang-Mills theory in terms of twisted reduced models. The model is interpreted as a manifestly star-gauge invariant lattice formulation of noncommutative gauge theory, which reduces to ordinary Wilson lattice gauge theory for particular choices of parameters. It possesses a continuum limit which maintains both finite spacetime volume and finite noncommutativity scale. We show how the matrix model may be used for studying the properties of noncommutative gauge theory.