Principal Investigator: Stephan Stolz
In previous joint work with Peter Teichner, the principal investigator was successful in collecting evidence that elliptic cohomology is closely related to super symmetric field theories by showing that the partition function of a super symmetric field theory of dimension 2 is an integral modular form; the space of super symmetric field theories of dimension 1 is homotopy equivalent to the infinite loop space determined by the real K-theory spectrum and that a 0-dimensional super symmetric field theory over a manifold represents an ordinary cohomology class (with real coefficients). These advances support the belief of the principal investigator that he and Peter Teichner have found the right way to incorporate `super symmetry' and `locality' in Graeme Segal's definition of quantum field theory. The principal investigator is optimistic that this might be the crucial step necessary to solve the two decade old problem of how to relate field theories and elliptic cohomology. A particularly interesting aspect is that the `push forward' maps in these cohomology theories seem to correspond to physicists' quantization procedures via functional integrals, which are mathematically not well understood.
The last two decades have seen very fruitful interactions between theoretical physics and topology. On a philosophical level this is not surprising since topology can be understood broadly as the study of `qualitative' aspects in mathematics, and it is to be expected that it is the QUALITATIVE aspects of a complicated quantum system (i.e., the aspects that don't change when varying the parameters of the theory) that should be easiest to analyze. This interaction has deeply influenced both areas with physicists using sophisticated tools from topology, and topologists incorporating ideas from quantum field theory. The principal investigator hopes that relating super symmetric quantum field theories of dimension 0,1 and 2 to well- studied objects in topology, namely `generalized cohomology theories' will significantly strengthen that interaction by bringing the calculational power of algebraic topology to physics and the physical/ geometric meaning of these cohomology classes to topology.