### Extended Quantum Field Theory and Cohomology, I

#### Posted by Urs Schreiber

A few minutes ago our little workshop on elliptic cohomology ended. All participants and speakers are on their way home.

I had the pleasure of helping organize this, attending it and giving a talk myself (on connections on $\mathrm{String}(n)$-2-bundles). While very gratifying, as a result I hardly found time for anything else. (I have to apologize to all those who are expecting comments to emails they sent me recently. I will try to provide these comments tomorrow or over the weekend.)

We had very few scheduled talks, the rest of the time being “discussion session”. The entire workshop was one big “russian style seminar”. And this was very fruitful, useful and effective. We shall do it this way again.

It turned out that several of the new ideas that our esteemed guests, Stephan Stolz and Peter Teichner, mentioned were closely related and had quite some overlap with things I was talking about in Toronto and some of the developments that have taken place since then: it’s all about how to understand various edges of the cube, which relates classical parallel $n$-transport with the corresponding extended $n$-dimensional quantum field theory and the relation of that to state-sum models like the FRS-formalism.

I am not entirely sure how much of the stuff which was discussed in our discussion sessions, much of which is unpublished and un-preprinted work, is supposed to be posted to a site like ours here. But what I shall do is talk about that stuff which overlapped with things I had thought about myself, and written about here before.

**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 $n \gt 0$.