18 - Topological gravity and algebraic topology

Summary

Introduction

This paper is an account, directed toward algebraic topologists, of Ed Witten's topological interpretation of some recent beakthroughs in what physicists call two-dimentional gravity; it is thus a paper about applications of algebraic topology. That algebraic topology indeed has applications was the thesis of Solomon Lefschetz's last book [20], which is an introduction to Feynman integrals. Today, the singularities which so concerned Lefschetz are avoided by string theory, which deals in a very clean and direct way with the space of moduli of Riemann surfaces; and I suspect that he would have been fascinated and delighted by the elegance of the problems which now interest mathematical physicists.

In a sense, the topic of this paper in the infinite loop-spectrum defined by Graeme Segal's category (with circles as objects, and bordered Riemann surfaces as morphisms, [2,33]); but by current standards, this is an immense object, related to the theory of Riemann surfaces as the algebraic K-spectrum is to linear algebra, and much current research is devoted to finding some aspect of this elephant susceptible to analysis by means of the investigator's favorite tool. In the present case, the subject will be the construction of topological field theories [3], using complex cobordism [1].

I first heard of these developments from a talk by Witten in the IAS topology seminar, and I learned more about them during subsequent conversations with him, with Segal, and with Peter Landweber.