Field theories, conformal nets and Kac-Moody groups

Summer school at the WWU Münster, Germany

October 8 - October 11, 2013

Organizers

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"

Schedule

Tuesday

Wednesday

Thursday

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