Modular Forms and Topology

We want to discuss various applications of modular forms in topology. The starting point is elliptic genus and its generalizations. The main techniques are the Atiyah-Singer index theorem, the Atiyah-Bott-Segal-Singer Lefschetz fixed point formula, Kac-Moody Lie algebras, modular forms and theta-functions. Just as the representations theory of classical Lie groups has close connections with the Atiyah-Singer index formula as exposed in [A1], the representation theory of loop groups plays very important role in our study. One of the most important new features of loop group representations is the modular invariance of the Kac-Weyl character formula, which allows us to derive many interesting new results and to unify many important old results in topology. In this paper we will develope along this line. We hope that the other features of loop group representations, such as fusion rules and tensor category structure may also be applied to topology. See the discussions in §3. The contents of this paper is organized in the following way. In §1 we introduce elliptic genus by combining index theory and the representation theory of loop groups. The relation of the classical index theory with representation theory of classical Lie groups was discussed in [A1]. Here one finds that just replacing the classical Lie groups by their corresponding loop groups, we get the complete theory of elliptic genus, or more generally the index theory on loop space. Especially the Dirac operator and the Witten genus of loop space are derived more convincingly in this way. Then all of the other well-known properties of elliptic genus, such as functional equations, characterizations by rigidity and fibrations, can be easily obtained. Here we only pick some less well-known results to discuss, for example the expressions of the parameters in elliptic genera in terms of theta-functions. Different from the classical Lie group case, a new feature in our situation appears, the modular invariance which, by combining with index theory, is applied to obtain many new topological results. This is the content of §2. Most results in this section are special cases of more general results. For simplicity we only give the main ideas of the proofs. We make some geometric constructions in §3 to understand elliptic cohomology. The construction in §3.3 is motivated by the vertex operator algebra construction of the monstrous moonshine module. In §3.1 we introduce vector