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.

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.

**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.

**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. Ran
spaces, factorization algebras, and factorization spaces. Applications to the
geometric Langlands programme.