# Spring 2014 MATH 276 001 LEC

Section | Days/Time | Location | Instructor | CCN |
---|---|---|---|---|

001 LEC | TuTh 2-330P | 35 EVANS | TEICHNER, P | 54453 |

Units/Credit | Final Exam Group | Enrollment |
---|---|---|

4 | NONE | Limit:25 Enrolled:8 Waitlist:0 Avail Seats:17 [on 05/28/14] |

**Lecture Course on Functorial Field Theories and Ring Spectra**

One important role of mathematics is to serve as a language for other sciences. There are many success stories, on the theoretical side general relativity (Lorentz geometry, Einstein’s equation) and quantum mechanics (functional analysis, Schrödinger’s equation) come to mind. However, new mathematical formalisms still need to be discovered for the physically relevant quantum field theories.

A successful approach, via **functorial field theories**, goes back to Witten, Atiyah and Segal. In this class, we will explain this approach from first principles and discuss several structural analogies to **ring spectra** studied in algebraic topology. For example, Sigma-models lead to spaces of field theories, deformations correspond to homotopies, central charge to the degree (or more generally, the twist) of a cohomology class. Gauge theories give equivariant cohomology classes and best of all, quantization is related to push-forward (or Gysin or wrong-way) maps. This is one way in which algebraic topology could contribute to making some of the ill-defined Feynman path integrals rigorous.

The class will consist of survey-like parts but we'll also explain those aspects in full detail that are most interesting to the participants. This will definitely include the theory of super manifolds because without super symmetry, the spaces of field theories are often contractible and hence don't give interesting ring spectra. We will also discuss a modern point of view on ring spectra via excisive functors, as well as symmetric, orthogonal ring spectra and Gamma-spaces.

In a seminar with Owen Gwilliams on **Factorization Algebras**, we'll be formalizing the perturbative aspect of quantum field theory. A close connection between this story and functorial field theories is currently under construction.

**Prerequisites:** Basic algebraic and differential topology: manifolds, vector bundles, homology and homotopy.

**Office Hours:** Tu 3:30-5:00, 703 Evans

**Recommended Reading:** Survey with Stephan Stolz: *Supersymmetric field theories and generalized cohomology*

Modern Ring Spectra: *Model categories of Diagram Spectra* by Mandell, May, Schwede, Shipley.

Physical background: *A pseudo-mathematical pseudo-review on 4d N = 2 **supersymmetric quantum field theorie*s by Yuji Tachikawa.

*Supersymmetry for Mathematicians: An Introduction* by V.S. Varadarajan.

**Course Webpage:** http://people.mpim-bonn.mpg.de/teichner/Math/FTRS.html