School
10 - 14 July 2023
Symplectic Singularities and Supersymmetric QFT Summer School
The goal of this summer school is to introduce the necessary background in algebraic geometry, representation theory, quantum field theory and string theory, with emphasis on the dialogue between the two communities, and to present the main results that have been obtained in recent years regarding symplectic singularities. Lectures will be given by physicists and mathematicians, and will be followed by exercise sessions. The target audience is PhD students, postdocs and established researchers interested in these topics.
There will be 4 lecture courses, 2 given by mathematicians for physicists, 2 given by physicists for mathematicians. Each course will consist of 4 lectures spread throughout the week.
Friday 14th of July is the French national holiday and the university buildings will be closed. On this day will organise a trip to the hortillonnages.
Lecturers
Gwyn Bellamy
University of Glasgow
Stefano Cremonesi
Durham University
Amihay Hanany
Imperial College
Alex Weekes
University of Saskatchewan
Abstracts of the lectures (4 hours each)
Gwyn Bellamy : Rational Cherednik algebras and Calogero-Moser spaces
The aim of these lectures is to give a gentle introduction to rational Cherednik algebras and associated Calogero-Moser spaces. I'll begin with the Calogero-Moser systems of n particles on the line and explain how this motivates the definition of the rational Cherednik algebra associated to the symmetric group. This leads naturally to the definition of rational Cherednik algebras associated to an arbitrary complex reflection group. These algebras have a quantized version (t = 1) and quasi-classical limit (t = 0). In the latter case t = 0, the Calogero-Moser (phase) space is realized as the spectrum of the centre of the algebra; we'll cover the basic properties of these spaces including the fact that they are isomorphic to Nakajima quiver varieties (Higgs branches associated to quiver gauge theory) for a large family of examples. Much of the work done by mathematicians has been to understand the representation theory of rational Cherednik algebras at t = 1 (quantized version). I'll give an overview of the representation theory of these algebras; in particular, introducing category O and its basic properties. Time permitting, we will see that rational Cherednik algebras are a subset of a larger class of algebras called symplectic reflection algebras. One can still associate a Calogero-Moser space to this larger class of algebras.
Stefano Cremonesi : Supersymmetric QFT and symplectic singularities
This course aims to introduce to mathematicians basic ideas of quantum field theory (QFT) and supersymmetry which underlie the appearance of symplectic singularities in theoretical physics. Topics will include: what is a QFT; what is supersymmetry; supersymmetric QFTs with 8 supercharges; how we study them.
Amihay Hanany : Branes and Quiver Gauge Theories
We will show how brane systems help in the construction of known and new symplectic singularities.
Alex Weekes : Symplectic singularities, Coulomb branches and affine Grassmannian slices
These lectures will begin with an overview of the basics of the theory of symplectic singularities. We will discuss links to Coulomb branches of 3d N=4 SUSY theories, as defined mathematically by Braverman-Finkelberg-Nakajima, before turning our focus to the special case of affine Grassmannian slices. These slices arise as Coulomb branches for quiver gauge theories (for finite type quivers), and their geometry has very interesting ties to representation theory. Finally, we'll discuss extensions (some conjectural) to general quivers.
Poster session & Wine and Cheese
Posters will be on display throughout the two weeks. There will be a dedicated Poster Session on Tuesday 11th, along with a Wine and Cheese degustation.
