Finding Generalized Cohomologies from Supersymmetric Field Theories

Abstract

Generalized Cohomology is a topic in Algebraic Topology. Field Theories are prominent in Theoretical Physics, with connections to the mathematical notions of Topological Field Theories. Supersymmetry can be added to the Field Theories. We will build bridges between these topics. We will introduce supermanifolds and stacks. Using those, we will define suitable bordism categories on which we define Supersymmetric Field Theories. We will see that the Field Theories are a geometric construction of some Generalized Cohomology Theories. We will construct ordinary cohomology from 0|1-dimensional Field Theories and complexified K-theory and complexified tmf from 1|1 and 2|1-dimensional Field Theories respectively. In these constructions, we aim to keep the dimensions general. In particular, we are able to relate even dimensional Field Theories to Siegel Modular forms.