Weekly blog - David Radnell

Friday 11.03.2016

posted Apr 5, 2016, 5:20 AM by David Radnell

This was part I of two lectures on the geometric definition of VOA. We mostly discussed background and motivation

Sketched G. Segal's definition of CFT.

Introduced the rigged moduli space of which there are two models: (1) Bordered Riemann surfaces where each boundary component is parametrized by S^1. (1) Punctured Riemann surfaces with local coordinate charts specified in a neighborhood of each puncture.

In higher-genus the complex structure of these moduli spaces requires serious work in Teichmuller theory which I have been working on for many years. We recently finished a review article: Quasiconformal Teichmuller theory as an analytical foundation for two dimensional conformal field theory

Stated the "Geometric definition of a VOA". (Which is based on the genus zero rigged moduli space). The sources for this are Yi-Zhi Huang's book and also his article in the Proc. Nat. Acad. Sci. "Geometric interpretation of vertex operator algebras".

Friday, 26.02 and 04.03

posted Mar 5, 2016, 7:20 AM by David Radnell [ updated Mar 5, 2016, 8:08 AM ]

Steven Flores gave a series of two lectures. He presented reconstruction theorems and used this approach to construct non-trivial example of vertex (operator) algebras. Detailed typed notes are in the "resources" section

Friday, 12/02/2016

posted Feb 13, 2016, 11:08 PM by David Radnell [ updated Feb 13, 2016, 11:51 PM ]

We continued with algebraic basics following Lepowsky and Li (pages 57 - 70, 84). In particular we discussed

Commutator and Associator forumlas
Weak commutativity, weak associativity and the D-derivative property. We sketched the proof of weak commutativity and outlined two of the key ingredients in proving the D-derivative property. Sugested HW: Fill in the details of these proofs.
Stated the theorem that in the definition of a vertex algebra, the Jacobi Identity can be replaced with "weak commutativity + D-derivative property".
Gave the definition of a VOA and discussed some basic properties of the conformal vector \omega and the formulas involving L(0), L(-1) and L(-2).
Very briefly discussed the restricted dual, "rationality of products" and the "commutativity" in the sense that of different expansions of a common rational function. This discussion was rushed at the end and I may have said something incorrect. Please see page 68, expression (3.2.4) and Proposition 3.2.7. Suggested HW: Look at the proof and fill in the details of this conceptually important proposition.
Highly recommended general reading: Terry Gannon, Vertex Operator Algebras. In the Princeton Guide to Mathematics. (A nice short 11 page overview of VOA)

Friday, 05/02/2016

posted Feb 5, 2016, 3:20 PM by David Radnell [ updated Feb 13, 2016, 10:50 PM ]

Following closely the presentation in Lepowsky and Li (see the resources page) we discussed the basics of the calculus of formal power series, the definition of a vertex alegbra and made some informal remarks about the axioms.

This material is in chapter 2 and and the first 5 pages of chapter 3. Suggested reading is everything up to and including page 53.

Working with formal series has its subtleties, especially the binomial expansion convention. Fill in the details of some proofs to get a feel for it. Another thing is to fill in some details of understanding the 3 variable delta function identities in terms of contour integration. More details appear in the appendix of Frenkel-Leposwky-Meurman. If somebody wants to present that it would be great. Let me know if interested.

I also suggest you read the introductions to all 4 books listed on the resources page.

Next week I plan to discuss (1) some of the axioms which are equivalent to the Jacobi identity (such as weak commutative and the D-derivative property), (2) the definition of vertex operator algebra, and (3) begin discussing the geometric notion of a VOA if time permits.