Field theories, conformal nets and Kac-Moody groups

Summer school at the WWU Münster, Germany

October 8 - October 11, 2013

Organizers

Alan Carey, Fabian Hebestreit, Michael Joachim, Daniel Kasprowski

The joint poster of the summer school and the conference "Twists, generalised cohomology and applications" is available here.

Program

There will be 3 lecture series (4 sessions per series):

André Henriques (Universiteit Utrecht, NL): "The string group via conformal nets"
Notes are available here (for the first three lectures) and here (for the last)!

The string group String(n) is the 3-connected cover of the orthogonal group O(n): it is an infinite dimensional topological group whose homotopy groups vanish in degrees < 4 and that it is equipped with a map to O(n) that is an isomorphism on homotopy groups above degree 3. Given a real vector space with inner product V, a string structure on V is a certain piece of extra data, call it S, with the property that the automorphism group of the pair (V,S) is the string group String(V).
The goal of this lecture series is to show that one can describe string structures using defects between conformal nets (these are the 1-morphisms in a certain 3-category).

Conformal nets are functors from the category of 1-manifolds to that of von Neumann algebras, and have their origins in a branch of mathematical physics called algebraic quantum field theory. The conformal nets that will be relevant for defining string structures are the so-called "free fermion" conformal nets, and are among the simplest ones to construct. There is a free fermion conformal net Fer(V) associated to every real vector space V as above. The main result that I will be presenting is that Fer(V)-Fer(R^n)-defects are string structures.

Nitu Kitchloo (Johns Hopkins University, Baltimore, USA): "Geometry and Topology of Kac-Moody groups"
Slides are available here!

Lecture 1 (The algebraic theory):
I will introduce various (equivalent) notions of loop groups. We will go into the Lie theory of algebraic loop groups by introducing the affine Lie algebra and the affine Weyl group. The example of the (untwisted) affine lie algebra of rank two will be studied in some detail. This corresponds to the group of loops on the compact Lie group SU(2).

Lectures 2,3 (The representation theory):
In these two lectures I will introduce an important class of representations of the loop group known as positive-enery representations. These decompose under a notion of level. We will classify all positive energy representations of a given level, and describe the Weyl-Kac character formula for irreducible representations. I will illustrate these formulas with examples. Our final goal will be to describe the geometric notion of fusion that induces a ring structure on the Grothendieck group of positive energy representations of a fixed level.

Lecture 4 (The homotopy theory):
In this lecture we will focus on the topology of the loop group and its classifying space. We will describe several structure theorems that pertain to the stable and unstable homotopy type of these spaces.

General remarks:
1) I will focus on loop groups since these are the groups most relevant to field theory, but I will point out how most of this theory extends to a larger class of groups known as Kac-Moody groups.
2) It will be helpful (though not necessary) if the audience is somewhat familiar with the theory of compact Lie groups and their classifying spaces.

Stephan Stolz (University of Notre Dame, Indiana, USA): "Field theories and cohomology"

Graeme Segal suggested two decades ago that families of 2-dimensional field theories parametrized by a manifold X should be related to a generalized cohomology theory of X, now known as "Topological Modular Form Theory". This is an analog of the well-known statement that homotopy classes of families of Fredholm operators parametrized by X can be identified with the K-theory of X.

In these lectures, based on joint work with Peter Teichner, I want to present a conjectural picture of TMF(X) as homotopy classes of families of supersymmetric 2-dimensional Euclidean field theories parametrized by X. Here the notion of a supersymmetric Euclidean field theory is a refinement of the well-known definition of a topological field theory as a functor out of a bordism category. I will present evidence for the conjecture in form of an analogous description of de Rham cohomology and K-theory of X in terms of supersymmetric Euclidean field theories of dimension 1 and 2, respectively. In addition I will outline our proof that the partition function of supersymmetric 2-dimensional Euclidean field theory is a modular form with integral Fourier coefficients.


Schedule

All lectures will take place in lecture hall M5, registration in room 505 (on the fifth floor of the tall building!) and the coffee breaks in room SR0. There will be signs showing you where to go.

Tuesday

9:00-9:30
Registration
9:30-10.30 André Henriques
11:00-12:00
Stephan Stolz
14:00-15:00
Question session
15:30-16:30
Nitu Kitchloo

Wednesday

9:30-10.30 André Henriques
11:00-12:00
Stephan Stolz
14:00-15:00
Question session
15:30-16:30
Nitu Kitchloo

Thursday

9:00-9:45 André Henriques
10:15-11:00
Stephan Stolz
11:30-12:15
Nitu Kitchloo
14:00-15:00
Question session
15:00 -
Excursion

Friday

9:00-9:45 André Henriques
10:15-11:00
Stephan Stolz
11:30-12:15
Nitu Kitchloo
14:00-15:00
Question session
15:00 -
Teatime

Registration and financial support

Registration is no longer available!

Location & how to get there

All lectures take place at the Mathematisches Institut - Westfälische Wilhelms-Universität in Münster (Germany). For a map click here. The nearest airport is Münster-Osnabrück (FMO). There is a bus (line S50) that takes about 40 min from the airport to the main train station of Münster. For train connections to Münster, check here; in the search form just write "ms" in the "Destination" field. This will determine the correct train station "Münster(Westf)Hbf". More information about transport to Münster can also be found here. For connections directly to our Mathematisches Institut please click here.

Accomodation

Participants are asked to make reservations themselves. If you have problems making a reservation or need additional information, please do not hesitate to contact us (email to Kirsten Sander). A list of suggested hotels (that may offer discounts upon mentioning the university of Münster) follows. The situation is a bit tight, since there is a congress here in Münster parallel to the summer school.
We have reserved a few double rooms (that can also be used as single rooms upon request) at Hotel Feldmann under the code "Summer school" until Sept. 7. Other standard options are:
Hotel Am Schlosspark
Hotel Bakenhof
Hotel Jellentrup
Hotel Busche am Dom
There is also a nice youth hostel (JugendGästehaus Aasee), which is about 2.4 km from the Mathematisches Institut.