During April-June 2020 IMPA will be hosting a series of activities focused on representation theory of vertex algebras and related topics. We will have a several minicourses, a graduate school and two workshops. For more information and registration instruction please refer to their respective pages in the menu above.
We begin by studying a toy model of the free boson from quantum field theory, and we see how the physical notion of renormalisation motivates the mathematical constructions of normal ordering, operator product expansion and ultimately vertex algebra. First example of vertex algebra: the free boson. Introduction to lambda bracket notation, and how to do computations in vertex algebras.
In this lecture we study a second example: the charged free fermion vertex algebra, by way of the Dirac sea construction (also from quantum field theory). We will see a nontrivial relation between the boson and the charged free fermion, which is connected with classical themes in number theory (the Jacobi triple product identity) and combinatorics (Schur polynomials).
We isolate the idea of the boson-fermion correspondence of the preceding lecture and extend it to yield a general construction of a vertex algebra associated with any integral lattice. The representation theory of these vertex algebras is much more beautiful than that of the boson, in particular there are only finitely many irreducible modules, which in a sense categorify the discriminant form of the lattice.
The Virasoro Lie algebra plays an important role in representation theory and mathematical physics. In this lecture we will more or less prove an important fact about it, called the Kac determinant formula. The simplest proof (that I know) uses vertex algebras, and is a simple case of an important strategy in representation theory known as "free field realisation".
In this lecture we look at some of the general axiomatics of vertex algebras, and see in particular how to get a Lie algebra from a vertex algebra. Applying this to the lattice vertex algebras gives a surprisingly simple uniform construction of the (simply laced) simple Lie algebras. Conversely we will see how to get a vertex algebra from a semisimple Lie algebra.
Arc spaces are of great importance in the theory of vertex algebras. One of the main reasons is that the sheaf of arcs over a scheme X has a structure sheaf of a commutative vertex algebras. Moreover, any vertex algebra is canonically filtered, and the associated graded space is a quotient of the space of functions on the arc space space of the associated scheme of the vertex algebra.
In my lecture series I will explain the above constructions in more detail. I will also present some applications of the use of arc spaces to vertex algebras. Lastly, I will also mention some open problems related to this topic raised by recent works of Arakawa, van Ekeren, Heluani and myself (among others)
In my lectures I will talk about a certain remarkable connection between vertex algebras and symplectic varieties arising from 4D/2D duality that was recently discovered by Beem, Lemos, Liendo, Peelaers, Rastelli and van Rees.
This school aimed at advanced graduate students and young researchers.
Registration: click here
Dates: May 25th 2020 to May 29 2020.
Confirmed minicourses by:
Registration click here
On leaving the airport we suggest that you take a taxi. There are several companies that have booths located inside the airport facilities, on the way out. There are also special buses leaving from Terminals 1 and 2, whose routes you can find here.
The bus line below has its final stop near the Institute:
You can check which bus line is best for you by clicking here.
Should you prefer to use a taxi, we suggest the following companies: