Yi-Zhi Huang

**Subtitle:** Intertwining operator algegras

### Text:

No textbook. Lectures are based on papers available online.

### Prerequisites:

First year graduate algebra and analysis courses. Basic knowledge in vertex operator algebras will be very helpful, but is not needed.

### Description:

Intertwining operator algebras are a class of algebras underlying a number of important algebraic and analytic structures in mathematics and physics. The main axioms of intertwining operator algebras are commutativity and associativity corresponding to operator product expansion and analytic extension property in physics. In the theory of vertex operator algebras and two-dimensional conformal field theory, the main objects to study are in fact intertwining operator algebras. They are equivalent to vertex tensor categories which in turn give braided tensor categories. They are precisely the algebras describing nonabelian anyons in physics. They are also exactly genus-zero chiral two-dimensional conformal field theories. When they are modular invariant, they also give genus-one chiral two-dimensional conformal field theories. They can be constructed naturally using the representation theory of vertex operator algebras.

In this course, I will introduce intertwining operator algebras and study the theory and applications of intertwining operator algebras without assuming that the students have any knowledge in vertex operator algebras or two-dimensional conformal field theory. Vertex operator algebras will be given as special examples of intertwining operator algebras. Below are the detailed topics to be covered in this course:

1. The representation theory of vertex operator algebras and a construction of intertwining operator algebras.

2. Examples of intertwining operator algebras from representations of affine Lie algebras and Virasoro algebras.

3. The construction of the moonshine module vertex operator algebra as an example of the construction of intertwining operator algebra.

4. Modular invariance of intertwining operator algebras.

5. Intertwining operator algebras, vertex tensor categories and braided tensor categories.

6. Intertwining operator algebras and nonabelian anyons.