Supersymmetric Field Theories and Cohomology

This is the Ph.D. dissertation of the author. The project has been motivated by the conjecture that the Hopkins-Miller tmf spectrum can be described in terms of ‘spaces’ of conformal field theories. In this dissertation, spaces of field theories are constructed as classifying spaces of categories whose objects are certain types of field theories. If such a category has a symmetric monoidal structure and its components form a group, by work of Segal, its classifying space is an infinite loop space and defines a cohomology theory. This has been carried out for two classes of field theories: (i) For each n ∈ Z, there is a category SEFT n whose objects are the Stolz-Teichner (1|1)-dimensional super Euclidean field theories of degree n. It is proved that the classifying space |SEFT n| represents the degree-n K or KO cohomology, depending on the coefficients of the field theories. (ii) For each n ∈ Z, there is a category AFT n whose objects are a kind of (2|1)-dimensional field theories called ‘annular field theories,’ defined using supergeometric versions of circles and annuli only. It is proved that the classifying space |AFT n| represents the degree-n elliptic cohomology associated with the Tate curve. To the author’s knowledge, this is the first time the definitions of low-dimensional supersymmetric field theories are given in full detail.