### Trimester Program on Vertex Algebras

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.

• Minicourses: April 6th 2020 - April 24th 2020
• School: May 25th 2020 - May 29th 2020
• Workshop: June 1st 2020 - June 5th 2020

### Jethro Van Ekeren: Introduction to vertex algebras (6 April 2020 -- 10 April 2020)

#### 1. The Boson Vertex Algebra.

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.

#### 2. The Fermion Vertex Algebra.

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).

#### 3. Lattice Vertex Algebras.

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.

#### 4. The Virasoro Algebra.

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".

#### 5. Affine Vertex Algebras.

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.

### Anne Moreau: Arc spaces and vertex algebras (13 April 2020 -- 17 April 2020)

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)

### Tomoyuki Arakawa: 4D/2D duality and representation theory (20 April 2020 -- 24 April 2020)

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.

### School on representation theory of vertex algebras

Location: IMPA

Dates: May 25th 2020 to May 29 2020.

Confirmed minicourses by:

• Tomoyuki Arakawa
• Terry Gannon
• Sergei Gukov
• Yi-Zhi Huang

### Workshop on representation theory of vertex algebras

Dates: 6/1/2020--6/5/2020.

Location: IMPA.

Confirmed Speakers:

• Tomoyuki Arakawa
• Katrina Barron
• Emily Cliff
• Chiara Damiolini
• Pavel Etingof
• Edward Frenkel
• Vyacheslav Futorny
• Azat Gainutdinov
• Terry Gannon
• Sergei Gukov
• Andrew Linshaw
• Sven Möller
• Anne Moreau
• David Ridout
• Vera Serganova
• Katrin Wendland

### Registered Participants

• Marco Aldi
• Iván Angiono
• Tomoyuki Arakawa
• Yusuke Arike
• Rodrigo Barbosa
• Katrina Barron
• Raphael Belliard
• Germán Alonso Benitez Monsalve
• Sandipan Bhattacherjee
• Dylan Butson
• Lucas Calixto
• Maria Clara Cardoso
• Anne Caroline Carneiro de Albuquerque
• Sungbong Chun
• Emily Cliff
• Giulio Codogni
• Chiara Damiolini
• Chongying Dong
• Pavel Etingof
• Justine Fasquel
• Zachary Fehily
• Edward Frenkel
• Azat Gainutdinov
• Terry Gannon
• Nathan Geer
• Naoki Genra
• Vassily Gorbounov
• Pinhas Grossman
• Marcela Guerrini Alvez
• Carlos Antonio Guimarães Silva
• Sergei Gukov
• Juan Guzmán
• Owen Gwilliam
• Xiao He
• Aron Heleodoro
• Oscar Armando Hernández Morales
• Yi-Zhi Huang
• Nguyen Anh Hung
• Aniket Joshi
• Julien Korinman
• Zongzhu Lin
• Andrew Linshaw
• Levgen Makedonskyi
• Robert McRae
• Ashish Mishra
• Yuto Moriwaki
• Kaveh Mousavand
• Sven Möller
• Shigenori Nakatsuka
• Gael Ngambali Ndzakima
• Henrique Oliveira Rocha
• Florencia Orosz Hunziker
• Kento Osuga
• Veronika Pedić
• Danilo Jose Polo Ojito
• Surya Raghavendran
• David Ridout
• Vera Serganova
• Fernando Junior Soares dos Santos
• Mathew Szczesny
• Juan José Villareal Montoya
• Katrin Wendland
• Hiroshi Yamauchi
• Nina Yu
• Wataru Yuasa
• Maxim Zabzine
• Hongbo Zhao
• André Silva de Oliveira

## How to get to IMPA

### From the airport

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.

### Buses

The bus line below has its final stop near the Institute:

• 409 - Saens Peña – Horto (via Lapa)

You can check which bus line is best for you by clicking here.

### Taxis

Should you prefer to use a taxi, we suggest the following companies: