From 1-matrix model to Kontsevich model

Loop equations of matrix models express the invariance of the models under field redefinitions. We use loop equations to prove that it is possible to define continuum times for the generic hermitian {1-matrix} model such that all correlation functions in the double scaling limit agree with the corresponding correlation functions of the Kontsevich model expressed in terms of kdV times. In addition the double scaling limit of the partition function of the hermitian matrix model agree with the $\tau$-function of the kdV hierarchy corresponding to the Kontsevich model (and not the square of the $\tau$-function) except for some complications at genus zero.Comment: 17 pages, Late

Quantum Gravity, Dynamical Triangulation and Higer Derivative Regularization

We consider a discrete model of euclidean quantum gravity in four dimensions based on a summation over random simplicial manifolds. The action used is the Einstein-Hilbert action plus an $R^2$-term. The phase diagram as a function of the bare coupling constants is studied in the search for a sensible continuum limit. For small values of the coupling constant of the $R^2$ term the model seems to belong to the same universality class as the model with pure Einstein-Hilbert action and exhibits the same phase transition. The order of the transition may be second or higher. The average curvature is positive at the phase transition, which makes it difficult to understand the possible scaling relations of the model.Comment: 27 pages (Latex), figures not included. Post script file containing 15 figures (1000 blocks) available from [email protected]

Review of AdS/CFT Integrability: An Overview

This is the introductory chapter of a review collection on integrability in the context of the AdS/CFT correspondence. In the collection we present an overview of the achievements and the status of this subject as of the year 2010.Comment: 31 pages, v2: reference added, references to other chapters updated, v3: footnote 1 on location of references added, v4: minor changes, references added, accepted for publication in Lett. Math. Phys, v5: minor corrections, links to chapters updated, attached IntAdS.pdf with all chapters in one file, see http://arxiv.org/src/1012.3982/anc/IntAdS.pdf or http://www.phys.ethz.ch/~nbeisert/IntAdS.pd

Review of AdS/CFT Integrability

This is the introductory chapter of a review collection on integrability in the context of the AdS/CFT correspondence. In the collection we present an overview of the achievements and the status of this subject as of the year 2010