nLab geometric model for elliptic cohomology

Idea

There are well-known geometric models for some cohomology theories. For instance

• integral cohomology is modeled by line bundles and bundle gerbes, etc.

and

• K-theory is modeled by vector bundles.

A geometric model for elliptic cohomology is supposed to be an analogous construction for elliptic cohomology or for tmf.

It is an old idea that analogous to how differential K-theory is modeled by parallel transport in vector bundles and hence by functors $Bord_1(X) \to Vect$, elliptic cohomology should somehow be modeled by functors $Bord_2(X) \to Vect$, where $Bord_2(X)$ is a 1-category of cobordisms equipped with maps to $X$.

Stephan Stolz and Peter Teichner have initiated a program studying this in the article What is an elliptic object?.

The fundamental idea of this program is essentially to encode parallel transport along cobordisms with maps into a given space $X$ pretty much along the lines of functorial differential cohomology?. One expects that this encodes actually the differential refinements of the corresponding cohomology theories, such as differential K-theory. However, currently in the program one divides out concordance which effectively divides out the differential information and keeps just the underlying topological information.

The fundamental result of this program so far is that there is a useful notion of 2d FQFT over $X$ such that its partition function is indeed topological modular form-valued, so that it is a candidate for a model of tmf.

The following is going to be an exposition of this partial result:

• Outline of the constructions and statements

  • Axiomatic field theories and their motivation from topology

  • (1,1)-dimensional Euclidean field theories and K-theory

  • (2,1)-dimensional Euclidean field theories and tmf

• Definitions

  • bordism categories following Stolz-Teichner

  • (2,1)-dimensional Euclidean field theories

See also the entry

• 2-spectral triple.

References

A recent survey of the program is

• Stephan Stolz, Peter Teichner, Supersymmetric Euclidean field theories and generalized cohomology (pdf)

The program was initiated in

• Stephan Stolz, Peter Teichner, What is an elliptic object?

After the original sketch in terms of extended FQFT subsequent work concentrated on ordinary 1-categorical constructions, the goal being to make very clear, and transparent and rigorous the constructions involved and the claim that

the partition function of a (2|1)-dimensional -Eudlidean FQFT is an integral modular form.

This latest development on this is in

• Stephan Stolz, Peter Teichner, Super symmetric field theories and integral modular forms (pdf)

warning I am being told that this is by now outdated and to be replaced by an improved version, which however is apparently not available yet