Topological Strings and Loop Equations

Abstract

The quantization of generally covariant field theories is a notorious difficult subject. Severe problems arise in the correct treatment of the dynamical metric that is introduced to achieve invariance under general coordinate transformations. Even in two-dimensions, where the issue seems much more accessible and essentially reduces to an understanding of the Liouville dynamics, an exact solution of quantum gravity has until recently remained elusive. However, as is demonstrated by this workshop, the application of an unconventional method, the realization of random surfaces by large N matrix integrals [1]–[3], has led to a remarkable breakthrough [4]–[8]. Soon after the double-scaling limit of the matrix model was solved, an alternative approach to two-dimensional gravity was suggested by Witten along the lines of topological field theory [9].