## October 25, 2006

### Hopkins Lecture on TFT: Introduction and Outlook

#### Posted by Urs Schreiber

I am currently in Göttingen, attending the lecture series by M. Hopkins that I mentioned recently #, called

Yesterday we heard the first of three parts, titled Introduction to topological field theories. Michael Hopkins mentioned a couple of basic notions and concepts of topological field theories and, in closing, briefly indicated some of the more profound issues concerning the relation between Chern-Simons theory, categorification, K-theory # and elliptic cohomology #.

Here is a transcript of notes I have taken during this first lecture.

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 $M$.

(1)$M \mapsto Z(M)$

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

(2)$Z(M_1 \sqcup M_2) = Z(M_1) + Z(M_2)$

and that it vanishes on boundaries

(3)$Z(\partial M) = 0 \,.$

(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,

(4)$\partial M = I \sqcup J^- \,,$

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

(5)$Z(I) = Z(J) \,.$

(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

(6)$\array{ I \stackrel{M}{\to} J \\ \; \downarrow f \\ 0 \stackrel{[0,1]}{\to} 1 }$

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

An example for such an invariant is the signature invariant

(7)$\sigma \,,$

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

(8)$q : H^*(M,\mathbb{R}) \times H^*(M,\mathbb{R}) \to \mathbb{Z} \subset \mathbb{R}$

on the real cohomology ring, defined by

(9)$q(x) = \frac{1}{2}\int_M x\wedge x \,.$

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

(10)$\sigma(M) = \int_M L(M)$

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

(11)$Z(M) = \int_M L(M) \,,$

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

(12)$Z(M_1 \sqcup M_2) \stackrel{\sim}{\to} Z(M_1) \otimes Z(M_2)$

and

(13)$Z(\partial M ) \to 1 \,.$

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:

(14)$\array{ d = 3 & \text{number} & ? & ? \\ d = 2 & \text{Vector space} & \text{K-class} & ? \\ d = 1 & modular tensor category & \text{K-module} & \text{tmf-class} \\ d = 0 & ? & \text{K-linear category} & \text{tmf-module} } \,.$

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.

Posted at October 25, 2006 9:46 AM UTC

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### Re: Hopkins Lecture on TFT: Introduction and Outlook

Maybe this is not the best place for this comment, but I hope you bear over with me. I am a mathematician with an interest in diffeomorphisms. I understand that one important property a QFT can have is diffeomorphism invariance.

When you study diffeomorphism type of manifolds and diffeomorphisms you run into a special class of cobordisms called h-cobordisms. These are cobordisms W from M to N (that is, the boundary of W is the disjoint union of M and N) with the very special property that the inclusions of M and N into W both are homology equivalences. In some cases this is enough to conclude that W is the product of M with an interval, and in particular that M and N are diffeomorphic. For instance, if M and N are simply connected, and the dimension of W is at least 6, this is so. This is the h-cobordism theorem of Smale. In dimension 4 and 5, things are more difficult. As an illustration, we do know that every 4-dimensional manifold homotopy equivalent to the 4-sphere is h-cobordant to the 4-sphere, but we do not know whether it is actually diffeomorphic to the 4-sphere.

Now to my question to you clever physics guys. Are there any signs that the concept of h-cobordism plays a role in physics? For instace, if two manifolds are related by an h-cobordism, would one expect some property of the corresponding induced map in some particular type of QFT?

Marcel Bökstedt

Posted by: marcel on January 27, 2007 10:48 PM | Permalink | Reply to this

### Re: Hopkins Lecture on TFT: Introduction and Outlook

That’s an interesting question. I’m sure there *must* be loads of stuff to say about how h-cobordism ideas relate to TQFT’s.

The best place I’ve seen to start thinking about this is the masterfully written exposition by Lupercio and Uribe on ‘Topological Quantum Field Theories, Strings and Orbifolds’. I’m specifically thinking of Example 1.0.2 on page 5 which shows how you get a 2d TQFT from the homology of a fixed manifold M.

Perhaps thats only a tenuous link between the ‘h’ in ‘h-cobordism’ and ‘TQFTs’, but I think its all part of a beautiful bigger picture which hasn’t yet been completely figured out. At least, if it has, no-one has told me :-).

For instance, Simon Willerton and Justin Roberts have done a lot of work on how you can get a TQFT from a complex manifold M - this is called Rozansky Witten theory. See eg. these notes.

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### Re: Hopkins Lecture on TFT: Introduction and Outlook

Wrt “these notes” and TQFT, you can recover any quantum invariant (qi) by substituting the right weight system in a Jacobi diagram. By applying Lie algebra weight system to universal diagram invariants you get all qi’s. For instance, by applying SU(2) weigth system to Kontesevich intergral you get the Jones polynomial.

Henry Delforn
Carpinteria, CA

Posted by: Henry Delforn on April 28, 2009 11:07 PM | Permalink | Reply to this

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