Witten Genus Seminar
Fall 2010
Max Planck Institute for Mathematics
We will not be meeting regularly until November. For the month of October, the relevant talks will be scattered amongst the Topics in Topology Seminar and Bruno Vallette's Operads and Deformation Quantization seminar at MPIM.In this seminar we will (attempt to) explain the paper by Kevin Costello, A Geometric Construction of the Witten Genus, II. The first paper is also a nice read, but doesn't include many of the very interesting details present in the second. There are two things we must grapple with in order to understand Kevin's construction: the formal derived geometry that he has developed for understanding perturbative QFTs, and the specific computations he carries out in his construction of holomorphic Chern-Simons theory.
The former topic will be complimented by Bruno's seminar, though we will have to add certain bells and whistles to expain curved L-infinity algebras, the form of Koszul duality used in the paper, and other related matters. However, my hope is that the focus of this seminar will be on Kevin's definition of a QFT, the specific example of holomorphic Chern-Simons, and how the Witten genus arises from this construction.
I would be remiss if I did not mention similar results. There is a paper by Gorbounov, Malikov and Schechtman on chiral algebras that is the starting point of this other story. Pokman Cheung has furthered this work, and constructs the Witten genus. His relevant papers are here and here.
Also, thanks to Owen Gwilliam and Ryan Grady for their useful comments.
Here are the notes from the talks.
On Costello's Definition of a Perturbative QFT
The definition used in the paper is formulated in a few places, notably Kevin's book on renormalization and effective field theory. He and Owen Gwilliam also have a very neat quantization theorem, which is in their developing wiki on factorization algebras and perturbative QFTs.List of Speakers and Talks
All talks will be in MPIM
Thursday October 14th, 10 am: Organizational Meeting and an Overview of the Paper, Daniel Berwick-Evans
Monday October 18th, 3 pm: The Cayley Plane and the Witten Genus, Carl McTague. Check out the paper.
Thursday October 28th, 10:30 am: Effective BV Theories, Arturo Prat-Waldron
Wednesday November 3rd, 4:30 pm: Examples of Effective BV Theories, Arturo Prat-Waldron
Wednesday November 10th, 4:30 pm: Curved L-Infinity Algebras, Joan Milles
Wednesday November 17th, 4:30 pm: Formal Geometry and the Atiyah Class, Carlo Rossi
Wednesday November 24th, 4:30 pm: Formal Geometry and the Atiyah Class II, Carlo Rossi
Friday December 3rd, 3:30 pm: Putting the Pieces Together: Holomorphic Chern-Simons Theory, Dmitri Pavlov
Wednesday Decmeber 8th, 4:30 pm: Counterterms and the Obstruction Complex, Qin Li
Friday December 10th, 3:30 pm: Finding the Witten Genus, Christian Blohmann