### 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.

Continue reading about lecture 1 here.

## 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).