Homology counts components and cycles, K-theory counts vector bundles and bundles of Clifford algebra modules. What about geometric models for other generalized cohomology theories? There is a vision, introduced by Segal, Stolz, and Teichner, that certain cohomology theories should be expressible in terms of topological field theories.
I will describe how the 0-th K-theory group can be formulated in terms of equivalence classes of 1-dimensional topological field theories. Then I will discuss what it means to twist a topological field theory, and explain that the n-th K-theory group comes from twisted 1-dimensional topological field theories.
The expectation is that 2-dimensional topological field theories should be analogously related to elliptic cohomology. I will take an extended digression to explain what elliptic cohomology is and why it is interesting. Then I will discuss 2-dimensional twisted field theory and explain how it leads us toward a notion of higher
("2-dimensional") algebra.
Based on joint work with Arthur Bartels and Andre Henriques