4-manifolds, quantum field theory and generalized cohomology

University Of California-Berkeley, Berkeley CA

Investigators

Abstract

In the first part of the project, the principal investigator will work on the topological classification of 4-dimensional manifolds, with an emphasis on infinite fundamental groups and the obstruction theory for embeddings of surfaces developed with Rob Schneiderman. The second part of the project is joint work with Stephan Stolz on the relation between generalized cohomology and super symmetric quantum field theories. For space-time dimensions d=0,1,2 there are now well-defined notions of (d|1)- dimensional Euclidean field theories over a manifold X. For d=0 these are provably closed differential forms and for d=1 one obtains K- theory of X by taking concordance classes. It is conjectured that the case d=2 leads to the long desired geometrical/physical interpretation of classes in the universal elliptic cohomology theory of "topological modular forms" of Hopkins, Miller and Lurie. Evidence for this conjecture comes from the recently proven fact that the partition function of a (2|1)-dimensional Euclidean field theory is an integral modular form. Both parts of the project relate methods from theoretical physics and mathematics. In the early 20th century, mathematical notions (like Riemannian geometry or functional analysis) were successfully used to explain physical theories (like relativity theory or quantum mechanics). In the second part of that century, the roles were somewhat reversed in that surprising predictions, mathematically provable only in very rare instances, were made by modern theoretical physics using notions like that of a quantum field theory. It is of ultimate importance for mathematical research to incorporate such notions into the body of well understood theories, hence providing both, progress in mathematics and a precise formulation of the basics of the physical theories. This project contributes to this goal via a particular relation between certain quantum field theories and some well understood cohomology theories in mathematics.