Intersection Theory, Integrable Hierarchies and Topological Field Theory

Abstract

The last two years have seen the emergence of a beautiful new subject in mathematical physics. It manages to combine a most exotic range of disciplines: two-dimensional quantum field theory, intersection theory on the moduli space of Riemann surfaces, integrable hierarchies, matrix integrals, random surfaces, and many more. The common denominator of all these fields is two-dimensional quantum gravity or, more general, low-dimensional string theory. Here the application of large-N techniques in matrix models, that are used to simulate fluctuating triangulated surfaces [1]–[3], has led to complete solvability [4]–[8]. (See e.g. the review papers [9], and also the lectures of S. Shenker in this volume.) Shortly after the onset of the remarkable developments in matrix models, Edward Witten presented compelling evidence for a relationship between random surfaces and the algebraic topology of moduli space [10, 11]. This proposal involved a particular quantum field theory, known as topological gravity [12], whose properties were further established in [13, 14] and generalized to the so-called multi-matrix models in [15]–[19]. This subject can, among many other things, be considered as a fruitful application of quantum field theory techniques to a particular problem in pure mathematics, and as such is a prime example of a much bigger program, also largely due to Witten, that has been taking shape in recent years. It is impossible to do fully justice to this subject within the confines of these lecture notes. I will however make an effort to indicate some of the more startling interconnections.