### Ben Zvi’s Lectures on Topological Field Theory II

#### Posted by John Baez

*guest post by Orit Davidovich and Alex Hoffnung*

Hi again! The following is a second set of notes which largely follows the second of David Ben-Zvi’s talks at a workshop on topological field theories, held at Northwestern University in May 2009. This post follows our previous post found here. We’ll again give a brief introduction, and then send you over to a PDF file for the full set of notes.

The first part of this lecture by David Ben-Zvi was dedicated to a discussion of the character and action maps that relate the open and closed sectors of our topological field theory. Attention is restricted to the case of representations of the form $A=\mathbb{C}[G/K]$ where $K\subset G$. In what follows ‘char’ and ‘act’ are defined via a pull-back and push-forward construction.

The second part of this lecture was dedicated to a discussion of topological field theories arising from gauge theory with symmetry group $G$, where $G$ is a complex reductive group. The first problem one encounters is that moduli spaces of fields on space-time manifolds, i.e., $G$-bundles with connections, are no longer discrete and do not allow for the same counting invariants that could be defined when $G$ was finite. This problem is solved by going up one dimension, namely, constructing $3$-dimensional TFTs.

One still encounters the issue of defining an appropriate theory of functions on moduli spaces of fields which are generally stacks. Following David, we discuss two natural categorified theories of ‘functions’. One is quasi-coherent sheaves and the other is $\mathcal{D}$-modules on $X$. Some of their properties that are essential for constructing the $3$d TFT are discussed.

Continue reading about lecture 2 here.

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

(The link to Alex's home page is wrong; it should be http://math.ucr.edu/~alex/.)