Courses | Invited talks | talks | 1 | 2 | 3 |

Abstract: Lecture 1: The bulk state space in conformal field theory
In this lecture I start by reviewing, as a motivation, various general results in CFT (two-dimensional conformal quantum field theory). In particular the relation between chiral CFT and full local CFT is made precise and some pertinent aspects of the so-called TFT-approach to rational CFT are described. Afterwards I discuss in detail algebraic properties of the space of bulk states of a CFT, as well as aspects of its character, which is the torus partition function.
Lecture 2: Categories and coends
In lecture 1 some informal knowledge about basic categorical notions is assumed. These as well as a few more advanced notions are now described in some detail. In particular the notions of ribbon category, modular tensor category, algebra and coalgebra objects, module of a monoidal category, finite tensor category, and factorizable finite tensor category are explained. The definition of the latter involves a certain Hopf algebra object $L$ (introduced by Majid and Lyubashenko), which can be constructed as a coend; accordingly the concept of a coend and related issues are also presented. Finally I discuss properties of the coend $L$ as well of another coend $F$ that turns out to be relevant in CFT. An important tool in the discussion is the graphical calculus for morphisms in strict monoidal categories.
Lecture 3: The Cardy-Cartan torus partition function
The categories of modules and bimodules over a finite-dimensional factorizable ribbon Hopf algebra $H$ are considered as models for the categories that arise in CFT. It is shown that the coend $F$ possesses all properties needed for the bulk state space of a CFT, and that the character of $F$ as a module over the coend $L$ for the $H$-bimodule category has the modular invariance properties needed for the torus partition function. A chiral decomposition of this character is described, which involves the Cartan matrix of the category of $H$-modules. Finally an outlook is given on how these results can be generalized to correlation functions of (not necessarily rational) CFT at higher genus.
Preparation material:
Appendices A and B.1-B.3 of hep-th/0503194
http://arxiv.org/abs/hep-th/0503194
Appendices of 1106.0210
http://arxiv.org/abs/1106.0210
Sections 2 and 4 and appendix A.1 of 1004.3405
http://arxiv.org/abs/1004.3405

Abstract: We begin by reviewing the blood and guts of conformal field theory: a tower of mapping class group representations, and with it the fusion rules, partition functions, etc.
We'll probably digress at some point and dabble in Mathieu Moonshine, just because it's there. Then I'll tell a story, about K-theory and CFT. I'll finish by explaining that subfactors predicts that there are many many well-behaved CFTs out there, waiting to be discovered!

Abstract: We will present operator algebraic approach to superconformal field theory.  It is a certain quantum field theory on the one-dimensional circle. We emphasize representation theoretic aspects, classification theory and connections to noncommutative geometry. Our method is functional analytic, and there is another object called a vertex operator algebra, which studies the same physical structure with an algebraic method.  We also make a comparison of the two approaches.