Conformal Field Theory and Elliptic Cohomology

In physics, conformal field theory is the basic step toward string theory. In mathematics, this concept has proven difficult to capture. Enormous progress was the discovery of vertex operator algebra (Borcherds, Frenkel, Lepowsky, Meurman, Griess, and others), but this concept does not give all the structure needed in physics, in particular where world sheets of higher genus are concerned. A more naive and farther reaching formalism was proposed by Segal and Kontsevich, but until recently, mathematical details of this approach had not been worked out. I will talk about this approach to mathematizing conformal field theory. As an application, I will show how a certain part of this method leads to progress in understanding elliptic cohomology.