The n-Category Café

I’ll reproduce my notes more or less verbatim the way I have taken them. Here and there I include additional personal comments, which are set in italics.

Topological Aspects of topological Field theory.

Topological field theory may be motivated by an interest in invariants $Z(M)$ of manifolds $M$.

On the left of this assignment we have a smooth manifold, possibly equipped with some extra structure, whereas on the right we have some “algebraic entity”.

In a theory of “classical” invariants one would demand that $Z$ is additive under disjoint union of manifolds

and that it vanishes on boundaries

(With an additional requirement on compatibility with direct product, this defines a genus)

This last condition, combined with the first one, is what justifies calling $Z$ an invariant: assuming that $M$ has two disconnected boundary components,

in other words, assuming that $M$ is a cobordism between $I$ and $J^ -$, then the above says that

(Here I am thinking of oriented manifolds and denoting orientation reversal by $(\cdot)^-$).

This says that $Z$ is invariant under cobordisms.

The first one to talk about this concept of cobordism invariant was apparently Pontryagin in the 1930s. While trying to work out the homotopy groups of higher spheres, he was thinking about maps from spheres to spheres, and in particular about the inverse image of regular points under these maps.

In particular, for our purposes it is convenient to think of a cobordisms $M$ with incoming boundary $I$ and outgoing boundary $J$ as equipped with a map to the interval

such that $f^{-1}(\{0\}) = I$ and $f^{-1}(\{1\}) = J$.

An example for such an invariant is the signature invariant

which assigns to each manifold $M$ the signature $\sigma(M)$ of the bilinear form

on the real cohomology ring, defined by

It is noteworthy that we can alternatively compute this signature as the integral

of a local quantity $L(M)$ defined on $M$, namely of a characteristic class of the tangent bundle $T M$. This is a consequence of the Hirzebruch signature theorem.

There is such a local integral version of every “classical” invariant of the above form, i.e. any cobordism inavriant $Z(M)$ can be written as

where the integral denotes the push-forward in some generalized cohomology theory $E$ and $L(M)$ denotes an $E$-valued characteristic class of the tangent bundle of $M$.

The slogan Michael Hopkins proposed for this situation was

The classical cobordism invariants of $M$ accumulate via integration of local linear approximations to $M$.

Our aim now is to pass from these “classical” inavriants to corresponding quantum invariants.

Like a classical cobordisms invariant is essentially a genus, a quantum invariant will be nothing but a topological field theory.

That is, for a quantum invariant we replace ring homomorphisms from the cobordism ring to some ring of numbers by a monoidal functor from the monoidal category of (diffeomorphism classes of) manifolds, to a monoidal target category.

This means that the two equations above are now replaced by morphisms

and

Here $1$ denotes the tensor unit of the target category. It is important that the first morphism above is required to be an ismorphism, while the second one is not.

In most of the following (and anyway in all of the remainder of this first part of the lecture) the monoidal target category is just that of finite dimensional vector spaces over some field.

Next, Michael Hopkins gave two simple examples of $(0+1)$-dimensional TFTs, one coming from oriented 1-dimensional manifolds (yielding “quantum cobordism invariants” of the oriented point) and one coming from 1-dimensional manifolds with spin structure (which is, despite its naive appearance, subtly different from the former case).

Maybe if I later find the time, I’ll spell out these two examples in more detail. Up to one issue that will apparently become important later (related to what Hopkins calls the flip map), they illustrate elementary points about TFT.

Right now, I shall skip these examples and jump to the concluding part of the first lecture, which provided an outlook on more sophisticated notions of TFT (often addressed as “extended” TFTs).

In this more sophisticated approach, people imagine not just assigning vector spaces to boundaries and linear maps to cobordisms, but to assign more generally, $n$-vector spaces to $(d-n)$-dimensional submanifolds (roughly), where $d$ is the top dimension involved.

(The first one to propose this picture was apparently Dan Freed)

For instance for 3-dimensional TFT (where apparently this is called the approach of three-tiered theories by (?) G. Segal), we would not just assign linear maps to 3-manifolds, vector spaces to 2-manifolds, but also modular tensor categories to 1-manifolds.

This can be found in Bakalov & Kirillov. Similar remarks can be found in the second part of Stolz&Teichner.

On his last slide, Michael Hopkins indicated that he has in mind (and is apparently going to partly explain in the next lectures) a more detailed picture for Chern-Simons $(2+1)$-dimensional TFT which includes information indicated in the following table:

Some of the question marks here are maybe already partly understood. But for our purposes they are question marks.

The step from the second to the third column of this table is supposed to be related to some kind of nerve realization or the like, which allows to go from categories to associated topological spaces.

Michael Hopkins promised that the step to the last column will be explained later.