Notes,
Witten Genus Seminar
 The primary goal of this seminar is to understand Kevin Costello's paper(s) on the Witten Genus. He constructs a QFT which recovers the Witten genus as its partition function. We will discuss necessary background and motivation along the way. Witten genus paper 1 Finished (arXiv link) Witten genus paper 2 Mostly finished. (see Costello's website also) Renormalization and the BV formalism The first appearance of the what appears in the book below. Kevin doesn't recommend reading it, but I highly recommend looking at it. Effective Field Theories Book This whole discussion takes place in the framework for quantum field theories laid out by Costello in this book. QFT-Costello-notes Handwritten course notes from a Quantum Field Theory course taught by Costello at Northwestern University (Winter 2009?) Fiorenza An Introduction the Batalin-Vilkovisky Formalism Berwick Evans Daniel Berwick Evans has notes from a seminar on Kevin's paper. The link on the left is the pdf of all notes from the seminar. Included are notes on points of difficulty. His site has more links to valuable resources. Caine (dvi) Notes on Gaussian Integrals. This is referenced in talk 3 below. A good place to see the fundamental computations which lead to the Feynman graph expansion formulas. Outline There is hopefully too much here. This is a rough plan of some things to cover in the early part of the seminar. I prefer to start out with too much and whittle down from there. If you are taking part in the seminar, please email my your comments. You can edit the source tex file, just email or talk to me, or, if you want to try to be technological, edit this google doc. Talk 1 - Grady Holomorphic Chern-Simons and the Witten Genus; Part 1: Stretching the Canvas. This is the first of a two part introduction to Kevin Costello's A geometric construction of the Witten genus, part 2. The main goal of these talks is to introduce the main players in the game and give the starting lineup; in future talks, the game will unfold. Additionally, we will touch on the relevance of the Witten genus and describe briefly the field theory paradigm in which Costello and others work Talk 2 - Grady Holomorphic Chern-Simons and the Witten Genus; Part 2. Combining effective field theories and the BV formalism. Main example: Chern-Simons. Extending Chern-Simons to manifolds of arbitrary dimension. Talk 3 - Thomas Quantizing the free scalar field theory. This follows the notes "QFT-Costello-Notes" linked above. The free scalar field theory serves as a guiding example with none of the difficulties we will encounter in other theories. (tex, booklet) Talk 4 - Thomas We add interacting terms to the action functional and prove The Feynman formula for the partition function. We formally define the category of (basic) graphs, where the objects are maps of finite sets and the morphisms are graphs. (tex,booklet) Talk 5 - Thomas We define counter-terms by way of example in the phi^4 theory. We introduce the notion of graphs with internal genus as a tool to keep track of the counter-terms. (tex) Talk 6 - Thomas Costello's definition of a perturbative quantum field theory with space of fields $C^\infty(M)$. The first main theorem states that any choice of a renormalization scheme gives a way to lift any Lagrangian (which one can think of as defining a classical theory) to a perturbative quantum field theory. (tex)