Operads and Conformal Field Theory Lecture 1: Conformal Field Theory

Physicists have developed very advanced methods to compute such integrals. For example, the famous Feynman diagram technique allows us to compute the above integral for a rather particular function S(x), a small perturbation of a positive de nite quadratic form in R . Sometimes, elaborate techniques of physicists to compute simple things nd an amazing application to very general things in mathematics, such as invariants of smooth four-manifolds. The purpose of this course is to give a mathematical introduction to two-dimensional quantum eld theory (2d QFT), which includes such theories as conformal eld theory (CFT), topological quantum eld theory (TQFT), string theory, quantum gravity, and cohomological eld theory. 2d QFT studies some particular objects related to Riemann surfaces (that is why 2d) or complex algebraic curves. These objects are close counterparts of the mathematical theory of operads, whose study is another purpose of the course. 2d QFT supplies ideas and motivation for such elds of mathematics as enumerative algebraic geometry, theory of moduli spaces, representation theory, low-dimensional topology, theory of knots, and a few others. That is why I believe it should be standard part of education of every serious mathematician. Theory of operads was created by algebraic topologists (J. P. May, J. M. Boardman, R. M. Vogt) in the seventies, but the very structure, later dubbed an \operad" by Peter May, had made its appearance in the sixties in the works of Jim Stashe on loop spaces and Murray Gerstenhaber on algebraic structures arising in deformation theory. Theory of operads is basically the theory of operations, as the name suggests, and no wonder, operations, such as the cup product, the Steenrod squares, the Dyer-Lashof operations, are an essential part of algebraic topology, whereas any product or bracket is what algebra is all about. Operads seemed to have played their role in the seventies, in the problem of recognizing a loop space in a given topological space, and since then has become in topology something like a steam engine. But in the nineties, the operads unexpectedly came back (Let me remind you that steam engines are not so harmful to the environment), allowing