Graduate lecture course, 8 lectures, Summer Term 2021.

Professor Joyce

Vertex Algebras

Tuesdays 2pm-3pm (or later), starting in week 1 (27th April) and finishing in week 8 (15th June), online via MS Teams.

This course is put on by the Taught Course Centre, and is available via MS Teams to graduate students (and staff, etc) at the Universities of Bath, Bristol, Oxford, Warwick and Imperial College London. I'm sorry, but random people from elsewhere can't come. If you are eligible, to sign up you should email gradstud@maths.ox.ac.uk with your MS Teams email address.

I will probably be casual about timing: I may not finish at 3pm if the material I have prepared takes longer than 1 hour, and I will encourage questions (without promising answers), so expect to finish between 3.0 and 3.30pm.

Note about broadening courses:

This course is too short to count as a broadening course for the purposes of satisfying EPSRC PhD regulations. Technically it could count as half a broadening course, but then you'd have to find another half-course to add up to a whole, and half-courses are usually not offered. So I suggest that you don't bother. If anyone asks, I will offer assessment as a (half) broadening course, by miniproject after the end of term. But if you are just looking for a broadening course to make the EPSRC happy, I recommend you do something else instead. It will probably be a terrible lecture course anyway, as I don't know anything about vertex algebras.

Bibliography:

My favourite books on vertex algebras:

V. Kac, Vertex Algebras for Beginners, University Lecture Series 10, A.M.S., 1997. (Short.)

E. Frenkel and D. Ben-Zvi, Vertex Algebras and Algebraic Curves, Mathematical Surveys and Monographs 88, A.M.S., 2001. (Longer, much more material.)

Some other useful books:

I. Frenkel, J. Lepowsky and A. Meurman, Vertex Operator Algebras and the Monster, Academic Press, 1988.

I. Frenkel, Y.-Z. Huang and J. Lepowsky, On Axiomatic Approaches to Vertex Operator Algebras and Modules, Mem. A.M.S. 104 no. 494, 1993.

J. Lepowsky and H. Li, Introduction to Vertex Operator Algebras and Their Representations, Progress in Mathematics 227, Birkhäuser, Boston, 2003.

Survey papers on vertex algebras:

C.A. Keller, Introduction to Vertex Operator Algebras, https://www.math.arizona.edu/~cakeller/VertexOperatorAlgebras.pdf.

C. Nozaradan, Introduction to Vertex Algebras, arxiv/0809.1380.

Synopses of lectures (work in progress!):

Lecture 1: Introduction. The Borcherds definition of vertex algebras. Vertex superalgebras, graded vertex algebras, and vertex operator algebras, defined in the style of Borcherds. Direct sums and tensor products of vertex algebras, quotient by ideals. Holomorphic vertex algebras, as first simple examples. (All other examples are complicated, see lecture 3 for more.) Constructing a (graded) Lie algebra from a (graded) vertex algebra.

Lecture 2: Formal power series notation. Defining vertex algebras in terms of formal power series. Identities satisfied in vertex algebras. Alternative axioms for vertex algebras. More motivation for vertex algebras, discussion of how vertex operator algebras arise in Physics.

Lecture 3: Locality, vertex operators, operator product expansions. Dong's Lemma. The Reconstruction Theorem. Examples of vertex algebras: Heisenberg vertex algebras, the Virasoro vertex algebra, vertex algebras from affine Lie algebras, and lattice vertex algebras,

Lecture 4: Representation theory of vertex algebras. Rational vertex (operator) algebras. Zhu's Theorem on modular invariance of characters of representations of cofinite rational VOAs. The Zhu algebra of an even graded vertex algebra V*, classification of simple V*-representations.

Lecture 5: More constructions of vertex algebras: the coset and BRST constructions, the Borcherds bicharacter constructions, orbifold vertex algebras.

Lecture 6: Monstrous moonshine: connections between the Monster simple group and modular functions, proved using vertex algebras. Infinite-dimensional Lie algebras and their representations. Kac-Moody algebras and generalized Kac-Moody algebras. Outline of Borcherds' proof of the Moonshine Conjectures.

Lecture 7: Interpreting vertex operator algebras as structures on all algebraic curves, following Frenkel and Ben-Zvi. Beilinson-Drinfeld's chiral algebras.

Lecture 8: Beilinson-Drinfeld's factorization algebras. Applications to the geometric Langlands programme.

PDF files to download:

Slides of lectures 1 and 2

Slides of lectures 3 and 4

Slides of lectures 5 and 6

Slides of lecture 7