Project III 2023-2024

Matrix models

Iñaki García Etxebarria

Description

There are a number of interesting problems in physics where one encounters integrals of the form \[ \int dM \, e^{f(M)} g(M) \] where \(M\) are some class of matrices (the set of hermitian matrices, for instance) and \(f(M)\) and \(g(M)\) are functions from matrices to \(\mathbb{R}\).

Physical models that reduce to integrals over such spaces of matrices are known as matrix models. These matrix models have a large number applications, ranging from polymer folding (relevant for chemistry or biology) all the way to topological string theory, or quantum gravity.

In this project you will learn about matrix models, and describe some of their applications to physics.

Group project

The group project will involve learning about the basics of matrix models, and their basic solution techniques. By the end of the group project you will have learned
  • What are matrix models
  • The orthogonal polynomials solution method
  • The saddle point approximation to matrix models in the one-cut case

Mode of Operation and Evidence of Learning for the group project

The project will revolve around learning through reading with focus on the underlying theory, mathematical rigour, and the development of deep conceptual understanding. Students will demonstrate their understanding by solving relevant problems, exploring examples and theoretical applications of the material, and clearly communicating it in both written and oral formats.

Individual project

The individual project will build on the knowledge we have gained in the group project. You could dig deeper into the solution techniques above, or you could look into more advanced applications of matrix models. A few examples of advanced topics you will be able to investigate are:
  • The relation between matrix models and quantum gravity in two dimensions
  • Applications to matrix models to the solution of supersymmetric Chern-Simons theory.
  • Folding problems in biology (for instance protein folding).

Mode of Operation and Evidence of Learning for the individual project

The project will revolve around learning through reading with focus on the underlying theory, mathematical rigour, and the development of deep conceptual understanding. Students will demonstrate their understanding by solving relevant problems, exploring examples and theoretical applications of the material, and clearly communicating it in both written and oral formats.

Pre-requisites and co-requisites

It is recommended, but not essential, that you should have taken Mathematical Physics II on the second year. If you don’t fulfill this pre-requisite but are still interested please talk to me.

Reading material