## July 8, 2009

### Ben-Zvi’s Lectures on Topological Field Theory I

#### Posted by John Baez

guest post by Orit Davidovich and Alex Hoffnung

Hi! What follows are some notes on the first of David Ben-Zvi’s talks at a workshop on topological field theories, held at Northwestern University in May 2009. We’ll start by sketching the basic ideas, and then we’ll send you over to a PDF file for more details and pretty pictures.

David Ben-Zvi devoted this lecture to $2$-dimensional gauge theory with finite gauge group $G$. The subject of finite group topological field theories started with Dijkgraaf and Witten in their paper Topological Gauge Theories and Group Cohomology. A gauge theory is a quantum field theory in which the fundamental fields are principal $G$-bundles with connection defined over a space-time manifold $M$. In classical field theory we restrict attention to connections satisfying equations of motion (e.g. flat connections). In quantum field theory we study the space of all connections by attaching $\mathbb{C}$-linear data to it. In topological field theory we study coarse features of the above depending only on the topology of $M$.

David’s goal for this lecture was to construct an extended $2$-dimensional topological field theory $\mathcal{Z}_G$. By ‘extended’ we mean to say that $\mathcal{Z}_G$ extends down to a point. It assigns ‘values’ to $2$, $1$ and $0$-dimensional manifolds. In what follows the path integral is used as a guiding principle.

Another approach to this construction would be the Cobordism Hypothesis which was first stated by Baez and Dolan in their paper on Higher-dimensional Algebra and Topological Quantum Field Theory. Lurie has announced a proof of a very general form of the Cobordism Hypothesis, which was the subject of his lectures at Northwestern, and which we will use in the later lectures. A sketch of the proof can be found in his paper On the Classification of Topological Field Theories. This would be done by showing $\mathbb{C}[G]$ is a fully dualizable object in a particular $2$-category. This would imply the existence of an extended $2$-dimensional TFT with $\mathcal{Z}_G(\bullet) := \mathrm{Rep}_\mathbb{C} G$, and allow us to compute its value for any manifold of dimension less or equal to $2$. This approach can be viewed as a ‘generators and relations’ type construction, while ours can be viewed as an a priori construction.

Posted at July 8, 2009 9:32 AM UTC

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### Re: Ben-Zvi’s Lectures on Topological Field Theory I

Many thanks to Orit and Alex for an amazing job with these notes!! (a marked improvement over the original..)

I’ve put my own badly-handwritten notes for the talks (mine and several others) at NW here . For Jacob’s beautiful lectures (or rather another iteration of them at UT Austin) there are videos and typed-up notes (by Braxton Collier, Parker Lowrey and Michael Williams).

Posted by: David Ben-Zvi on July 8, 2009 2:59 PM | Permalink | Reply to this

### Re: Ben-Zvi’s Lectures on Topological Field Theory I

Thanks for posting these various talks. What is known about the categorification of Frobenius algebras?

For instance, the earlier work of Kevin Costello shows that a Frobenius algebra is a Calabi-Yau category with one object. Also, a commutative Frobenius algebra occurs in 2Cob, but is there also anything else that I am missing?

Posted by: Charlie Stromeyer on July 9, 2009 1:29 PM | Permalink | Reply to this

### Re: Ben-Zvi’s Lectures on Topological Field Theory I

If you examine Kevin Costello’s paper you’ll see that a 1-object Calabi–Yau $A_\infty$ category is an infinitely categorified version of a Frobenius algebra — see Section 2.

And as we’ve discussed here, a modular tensor category is an example of a ‘once categorified’ Frobenius algebra.

However, the former example doesn’t include the latter — unless there are tricks that I don’t understand, which is entirely possible. The reason is that an $A_\infty$ algebra is a kind of weak monoid in the $(\infty,1)$-category of chain complexes, while a tensor category is a kind of weak monoid in the 2-category of Kapranov–Voevodsky 2-vector spaces.

If you’re interested in categorified Frobenius algebras, you should also check out week268, including the discussion here on the $n$-Café. You’ll see that categorified Frobenius algebras are important in propositional logic and the study of ‘thick planar tangles’, which look like this:

All these applications of categorified Frobenius algebras are deeply related, but the applications to logic have not been integrated with the rest.

Posted by: John Baez on July 9, 2009 2:02 PM | Permalink | Reply to this

### Re: Ben-Zvi’s Lectures on Topological Field Theory I

Thanks, John. This is interesting, and another thing that’s cool and analogous is the work of Benjamin Cooper which shows that an L-infinity algebra gives in a functorial way a certain type of 3d TFT:

http://www.math.ucsd.edu/~bjcooper/research.pdf

Posted by: Charlie Stromeyer on July 9, 2009 2:42 PM | Permalink | Reply to this

### Re: Ben-Zvi’s Lectures on Topological Field Theory I

Charlie - There are two different roles you mention for Frobenius algebras in 2d TFT: commutative Frobenius algebras arise as the values on the circle, while (not.nec.commutative) Frobenius algebras arise as generators for the category of “branes”, i.e. what an extended TFT assigns to a point or an open-closed TFT has as labels. By “generator” I mean you describe the category in terms of modules for this algebra.

So there are different ways to categorify each. For the case of the circle (the commutative case), we can pass to modular tensor categories as John explains - they arise from extended 3d TFTs (or categorified 2d TFTs) on the circle. We can also “homotopify”, which can be considered a kind of categorification, and replace commutative Frobenius algebras in vector spaces by an analog in chain complexes - this is what arises on the circle in Costello’s TCFTs, e.g. in string topology and the A- and B-models of topological strings.

At the level of a point, a 1-object Calabi-Yau category is (essentially by definition) a Frobenius algebra, while the A_oo version that appears in Costello (as John explains) is a homotopical analog thereof. Any reasonable CY category can be thought of as modules for such a homotopical Frobenius algebra.

But we can also categorify this picture, by looking at what a categorified 2d TFT - or a 3d TFT - assigns to the point.. this is now some kind of 2- (or $(\infty,2)$-) category, and we can describe it in terms of a different categorified notion of Frobenius algebra (or eg if it has one object it will BE such a notion).

An example of this is studied in detail in my paper with David Nadler on Character Theory - we show that the “finite Hecke category”, a categorified analog of the group algebra of the Weyl group (or Artin braid group) of a semisimple Lie group, is such a beast, a categorified (or really $(\infty,2)$-categorified) Frobenius algebra. Its center (defined appropriately) is what you’d assign to the circle in this TFT, and thus a homotopical analog of a modular tensor category (or categorified commutative Frobenius algebra). It’s a categorified analog of the (commutative Frob. algebra of) class functions on a group, which are known as character sheaves.

Posted by: David Ben-Zvi on July 9, 2009 3:04 PM | Permalink | Reply to this

### Re: Ben-Zvi’s Lectures on Topological Field Theory I

Cool stuff. I promise to read this paper sometime… sorry to be so slow about it!

Posted by: John Baez on July 14, 2009 6:13 PM | Permalink | Reply to this
Read the post Ben Zvi's Lectures on Topological Field Theory II
Weblog: The n-Category Café
Excerpt: This is the second set of notes on David Ben-Zvi's lectures on topological field theory at Northwestern University.
Tracked: July 14, 2009 6:11 PM

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