Mathematical Sciences: Two-dimensional Topological Field Theories and Complex Cobordism

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Under support from DMS 8713718, a problem of paramount interest was to generalize elliptic cohomology so as to encompass curves of higher genus. Y. Shimizu and the investigator found a solution to this problem (over the rational numbers), based on the idea that affine connections on elliptic curves (which do not generalize to higher genus) should be interpreted as special cases of the projective connections (which exist for all Riemann surfaces). This allowed them to associate in a natural way to any chiral conformal field theory (in the sense of Segal), the topological genus defined by the projective connection (or stress-energy tensor) of the field theory. They were able to identify this genus in the case of the charge-one fermion and to show that it reduced (in the case of the elliptic curve) to the elliptic genus. A corollary of this construction is that the complex cobordism ring supports a canonical quadratic differential, closely related in some way to a quadratic differential supported by the fundamental bosonic representation of the Virasoro algebra. The relation between these two quadratic differentials seems to be of fundamental importance and will be a major emphasis of the current project. In recent years mathematical constructions inspired by theories used to make sense of subatomic particles have found repeated application to purely geometric problems. To their mutual profit physicists and mathematicians are interacting more than ever in exploring these connections, and the investigator is a major player in this game.