Lectures on “Free field methods in Mathematical Physics and Representation Theory”

From Monday 24th July 2023
To Friday 20th October 2023
Location Zoom, and Evan Williams Theater (Peter Hall), Melbourne University

Research fellow Jean-Emile Bourgine will be giving series of lectures on free field methods at Melbourne Uni, also available by Zoom.  Interested people, including especially graduate students, are welcome to contact Jean-Emile for the Zoom link.  Details are below:

Lectures on “Free field methods in Mathematical Physics and
Representation Theory”
by Jean-Emile Bourgine

Every Friday 2:15~3:15 in Evan Williams Theater (Peter Hall)
Zoom link will be sent by email, if you are interested please contact
me ( jean-emile.bourgine@unimelb.edu.au ) to be added to the mailing

Description: The goal of this series of lectures is to present a set of
algebraic techniques inspired by Quantum Field Theory and based on the
notions of free fields and vertex operators. These techniques can be
applied to a wide range of mathematical problems in representation
theory (free field representations), quantum systems and lattice models
(quantum groups, Jimbo-Miwa vertex operators), symmetric functions
(Hall-Littlewood and Macdonald polynomials) and differential equations
(integrable hierarchies). Besides, these lecture will give an overview
of different important research topics in mathematical-physics, and can
serve as an introduction to essentials tools of quantum field theory
and string theory.

Plan (temptative):
1 The quantum harmonic oscillator
1.1 Importance of harmonic oscillators in physics
1.2 Quantization and the Schrodinger equation
1.3 Ladder operators and Fock states
1.4 The bracket notation
1.5 Toward vertex operators

2 Free bosons
2.1 Classical and quantum free fields
2.2 Heisenberg algebra and bosonic Fock space
2.3 Symmetric functions
2.4 Vertex operators

3 Free fermions and Schur functions
3.1 Dirac fermions
3.2 Fermionic Fock space
3.3 Boson-fermion correspondence
3.4 Schur functions

4 t-fermions and Hall-Littlewood symmetric functions
4.1 t-fermions algebraic relations
4.2 Hall-Littlewood polynomials

5 Integrable hierarchies of differential equations
5.1 W1+∞ and gl(∞) symmetry
5.2 Tau functions and Hirota equation
5.3 From Hirota to KP

6 (quantum) W-algebras
6.1 Virasoro algebra
6.2 W-algebras

7 Introduction to quantum groups
7.1 Hopf algebras
7.2 R-matrix and Yang-Baxter equation
7.3 Quantum affine sl(2) algebra

— if time permits —

8 The Jimbo-Miwa vertex operators
8.1 Six vertex model
8.2 Vertex operators as intertwining operators
8.3 Calculation of correlation functions

9 Quantum toroidal gl(1) algebra and Macdonald polynomials
9.1 Vertical/Horizontal Fock representations
9.2 Application to Macdonald polynomials