The String Coffee Table

The talk had two parts:

I Review: Field Theories

II Outlook: Field theory objects

I Review. Euclidean $(d=1)$ Field Theories

For 1-dimensional euclidean field theories the main idea is this.

A 1-dimensional euclidean field theory is a functor from enriched 1-dimensional cobordisms to Hilbert spaces. In physics language, this can be thought of as modeling propagation in quantum mechanics. The bordisms model an abstract “non-relativivistic” worldline and the functor is the propagator.

In order to get something interesting related to cohomology from such a functor, there needs to be certain extra structure. Crucially, we need to implement a certain grading and we need to have supersymmetry. The grading will be implemented by having a Clifford algebra represented on our Hilbert spaces and making the functor Clifford linear in a certain sense. (With a little luck I find the time to spell out technical details in another post.) This will make us want to enrich our bordisms by certain Clifford data. The result is a

Clifford linear 1D EFT of degree $n$

This is not quite enough. We want the bordisms to carry the structure of superspaces and want the functor to be a morphism of supermanifolds. Implementing this yields a functor

When properly set up, the main result is the following.

- The space of all such 1D super euclidean field theories at grade $n$ is homotopy equivalent to the $n$th space in the spectrum for (real) K-theory.

- There is a map from connected components of the space of all such field theories to the real K-theory of a point ($\simeq \mathbb{Z}$), called the partition function.

In the case where our functor is the propagator obtained from a Hamiltonian arising as the square of a Dirac operator on some Riemannian target space, this partition function is, well known, the index of that Dirac operator, constituting the so-called $\hat A$-genus of the target manifold.

We want to generalize this setup to 2-dimensional field theory.

Before going into this, I shall make a personal remark on one possibly helpful way to think about this. The rest of this entry is my point of view, not that expressed in the talk that I really wanted to talk about… (to be continued in the next entry.)

Why is it that the space of all suitably formulated supersymmetric quantum mechanical systems is what is called the classifying space for K-theory?

Think of it this way.

Assume you have a supersymmetric quantum mechanical system with a time independent Hamiltonian and supercharge. I will address this supercharge in general as a Dirac operator. Of course, only in special cases will it be the true Dirac operator of some ordinary Riemannian space. More generally it might be the Dirac operator on some noncommutative geometry, namely if Alain Connes’ axioms happen to be met by it. In even more general cases we may allow ourselves to think of any single odd-graded operator which defines a functor $\mathrm{SEFT}_n$ as some even more generalized notion of Dirac operator/geometry.

All right, so in this case our propagator is essentially given by the familar formula for (imaginary-time) quantum mechanical propagators

The operator $D$ has a kernel and a cokernel - two vector spaces. The partition function is the supertrace of the above propagator and computes the index of $D$, which is the difference between the dimension of the kernel and the cokernel. We should, however, think of this integer not just as a number. Given the kernel and the cokernel, we should regard the formal difference vector space

This is a virtual vector bundle over a point. Hence a representative of a class in K-theory of the point.

Therefore, one sometimes hears or reads people concerned with these issues saying that the physical interpretation of all this just works for K-theory over a point.

But I think that’s not the right way to think about it. In some sense at the level of precision of the present discussion I claim that the K-theory of spaces consisting of more than just one point is given by time dependent supersymmetric quantum mechanics.

Namely, in general our 1-dimensional bordisms come equipped with a function $\phi$, that maps them into some space $X$. But this function has to be compatible with the composition of bordisms. Hence it is really a functor from abstract bordisms to paths in $X$. The discussion above is the case where this space $X$ is a point and we can forget about this map $\phi$.

But by the reasoning familiar from the theory of parallel transport - or, equivalently, by the reasoning involved in what in QM textbooks is called the Dyson series - the above result on the form of our $\mathrm{SEFT}_n$-functor for $X = \mathrm{pt}$ tell us how the $\mathrm{SEFT}_n$-functors behave for nontrivial $\phi$. Namely we get propagators as in time-dependent quantum mechanics:

Here $\mathrm{P}$ denotes path ordering, as usual.

One sees here that $\phi$ together with our functor $\mathrm{SEFT}$_n encodes how the Hamiltonian and supercharge/Dirac operator of our system trace out a path in some family of operators which is parameterized by $X$ as wordline time (on the abstract bordisms) passes by.

How does this give rise to the K-theory of $X$? (Thanks again to Guy Buss for pointing out the following fact to me.)

At each time $t$, for each value of $\phi(x(t))$ we obtain a fixed Dirac operator $D(\phi(x(t)))$. This has a kernel and a cokernel. As we continuously walk through the space of Dirac operator the difference of the dimension of each of these two spaces seperately may jump, but their differece is a constant. Hence we can form again a well defined formal difference vector space $\mathrm{ker}(D(\phi(x(t)))) - \mathrm{coker}(D(\phi(x(t))))$. This now lives over the given point of $X$. Doing so for each point of $X$ (i.e. for each Dirac operator that may appear at one instant of time as the supercharge of our QM system) we obtain a virtual vector bundle over $X$!

This virtual vector bundle defines an element in the K-theory of $X$. Hence, from this physical point of view, the fact that the K-theory of a space is computed by maps from that space into a space of Fredholm operators is interpreted such that this map gives a way to encode the time-dependence of a Dirac operator of a time-dependent supersymmetric quantum mechanics.

Let’s try to generalize this to 2-dimensional field theory.

Roughly, the idea is now the following. Our (superconformal) 2-dimensional field theory is now a 2-functor from little pieces of euclidean string (to be thought of as a “non-relativistic” string, like we have “non-relativistic” quantum mechanics before).

Again, these pieces of worldsheet may be equipped with a function (really a 2-functor) $\phi$ which maps them into some space $X$. If $X$ is a point, our 2-functor

will be given by a choice of super-Virasoro generators $L, \bar L, D$. Here $L + \bar L := H$ is to be thought of as the worldsheet Hamiltonian and $P = L - \bar L$ as the generator of “string rotations”. In order to get something interesting we want only one of these two generators to have a supersymmetric square, hence

Stolz and Teichner call this $1/2$-supersymmetry. But of course to physicists this is known as $(1,0)$ supersymmetry - the heterotic string.

(When I pointed this out to several mathematicians at our workshop most of them laughed and found this terminology rather insinuating. :-)

So our 2-functor here is somehow given by a propagator formula as

Now, as before, we can compute the partition function of this operator. This is now the generalization of the index of an ordinary Dirac operator, which yielded the so-called $\hat A$-genus. But now this number is called the Witten genus. Since $D$ is roughly a Dirac operator on loop space, you can roughly think of this as the index of a Dirac operator on loop space.

The important point is that, as we have seen before, these genera are to be thought of not simply as numbers. They are really elements of some cohomology of a point. In the previous exammple the index was really a decategorification of a vector bundle over a point.

Here, similarly, we should expect the Witten genus to really arise from the decategorification of a virtual 2-vector space, arising as the formal difference of the 2-kernel and the 2-cokernel of the 2-Dirac operator encoded by the $D$ above.

Now consider the physically somewhat unusual situation where our super Virasoro generators are not fixed, but may vary from point to point on the worldsheet. This is what is now encoded by equipping our worldsheet with a function $\phi$ into some space $X$ and by applying our 2-propagation (“2-transport”) 2-functor to that enriched 2-bordism 2-category. By an analogous reasoning as above, this now encodes sort of a 2-dimensional time dependence of our superconformal theory. In particular, it associates the pre-decategorification kernel-cokernel data of the “time”-dependent loop space Dirac operator to each point of $X$. Hence such a “time-dependent” transport 2-functor should define some sort of virtual 2-vector bundle over $X$.

And this virtual 2-vector bundle should represent a class in what is called

This story about our 2-functor associating 2-vector bundles to $X$ is an observation on my part, intended to make transparent what one is expecting to see. Namely, people working on this are expecting that the space of 2-dimensional superconformal euclidean field theories at grade n is a classifying space for elliptic cohomology, or, rather, for the spectrum called

I have to run now. To be hopefully continued when there is more time…