The n-Category Café

Reminder: Stolz-Teichner’s program of realizing Generalized Cohomology in terms of Quantum Field Theory

I once tried to write a summary of the ideas and concepts relevant here in Seminar on 2-Vector Bundles and Elliptic Cohomology, V. You can also find lots of references assembled there and in Seminar on 2-Vector Bundles and Elliptic Cohomology, I.

Some aspects of the more recent developments can be seen in the preprint

S. Stolz & P. Teichner
Super symmetric field theories and integral modular forms

It is (by now a decade-old) idea, dating back to G. Segal and later picked up greatly developed by Stephan Stolz and Peter Teichner, that generalized cohomology theories, which are functors $$H^n : \mathrm{Spaces} \to \mathrm{AbGrp}$$ should have a “geometric realization” in terms of quantum field theories, i.e. in terms of representation categories of cobordisms “over spaces”.

So, roughly, if, for $X$ any space, $$d\mathrm{Cob}(X)$$ denotes some notion of (ultimately $d$-extended) category of $n$-cobordisms $\Sigma$ which are equipped with (suitably well behaved) maps $$f: \Sigma \to X$$ to the space $X$, and if $$d\mathrm{Vect}$$ denotes some useful flavor of the (ultimately $n$-)category of vector spaces, then the functor

$$X \mapsto \mathrm{Hom}(d\mathrm{Cob}(X),d\mathrm{Vect})_\sim$$

should yield a generalized cohomology theory, at least if some suitable bells and whistles are added to this.

Here the notation ${}_\sim$ denotes identifying what are (by now at least) called concordance classes of field theories. This can roughly be thought of as identifying field theories which may be related by continuous deformations.

Quick plausibility argument.

As a quick heuristic argument why this is a reasonable program, it pays to look at the simple cases of degree 0-cohomology.

i) – deRham cohomology —

The simplest example is ordinary deRham cohomology. $H^0(X)$ here is the additive group of constant functions on $X$.

On the other hand, 0-dimensional quantum field theory parameterized by $X$ is a 0-functor $$0\mathrm{Cob}(X) \to 0\mathrm{Vect} \,.$$ Since $0\mathrm{Vect}$ is nothing but our ground field that’s nothing but a function on $X$. Doing this carefully and dividing out concordance classes produces the collection of all constant functions on $X$, indeed.

ii) – K-theory —

The next simple case is KO-theory $KO^0(X)$. This is just the Grothendieck group of vector bundles on $X$.

On the other hand, 1-dimensiona quantum field theories parameteried by $X$ are 1–functors $$1\mathrm{Cob}(X) \to \mathrm{Vect} \,.$$ These assign in oparticular a vector space to every point of $X$. If done carefully and correctly, with all the right bells and whistles added, the collection of concordance classes of such functors is indeed the Grothendieck group of vector bundles over $X$.

iii) – something like elliptic cohomology —

This pattern continues. In degree 0 the cohomology theories we get from $d$-dimensional extended quantum field theories this way essentially assign to each space $X$ the (Grothendieck group of) $d$-vector bundles on $X$.

It is another old conjecture, which we talk about every once in a while here, that something like elliptic cohomology should come out by replacing, in K-theory, vector bundles by 2-vector bundles.

This is one way to see why people expect that the space of (certain) 2-dimensional quantum field theories, “parameterized by a space $X$”, knows about the elliptic cohomology of $X$.

The major two bells-and-whistles that needs to be added to this setup are

a) smoothness: everything has to depend smoothly on all the data

b) supersymmetry: all spaces and cobordisms involved should be supermanifolds (hence the filed theories be “super quantum field theories”). Only this makes the cohomology theories obtained from assigning field theories to spaces nontrivial. Moreover, this is required to get the cohomology theories $H^n$ for $n \gt 0$.