Project IV (MATH4072) 2023-24

The elliptic genus of K3 and Mathieu Moonshine

An elliptic genus is a special type of genus developed as a tool for dealing with questions related to quantum field theory. For a compact complex manifold, one can define its elliptic genus as a function in two complex variables. If the first Chern class of the complex manifold vanishes, then the elliptic genus is a Jacobi modular form with integral Fourier coefficients. There exist different techniques to calculate the elliptic genus of such manifolds. In this project we are interested in the elliptic genus of a K3 surface, and how it can be calculated using tools from superconformal field theory, which naturally stems from considerations in string theory. This approach offers a light on elliptic genera that is completely different from that provided by geometric methods. It turns out that if the string theory ‘knows’ about a K3 surface, the calculation of the elliptic genus of K3 via conformal field theoretic techniques reveals the presence of a huge finite group called ‘Mathieu 24’ and to this day, nobody really knows the role of this group in string theory. This phenomenon has been coined ‘Mathieu Moonshine’.

The goal of the project is to calculate the elliptic genus of K3 using superconformal characters as building blocks of the construction. One construction is the so-called ‘Gepner method’, another uses orbifold conformal field theory. All such theories lead to the same elliptic genus, which is a topological invariant, so calculating it from different perspectives would allow to check this property. Typically, one would calculate the elliptic genus for some Gepner models and/or for ${\mathbb{Z}}_2$ and ${\mathbb{Z}}_3$ orbifold conformal field theories.

A precise definition of elliptic genus of K3, of modular form, of $N=2$ and $N=4$ superconformal algebras, their representations and characters will be given to kick-start the project, which can evolve in a number of directions.

Prerequisites

The project requires a good command of Complex Analysis II (MATH2011) and taking Topics in Algebra and Geometry IV (MATH4151) as a corequisite may be helpful as it introduces modular forms.

Resources

Read What is an elliptic genus? by Serge Ochanine for an idea of the geometry behind the elliptic genus

The Mathieu Moonshine phenomenon was first observed in the article Notes on the K3 surface and the Mathieu group 24 by Eguchi, Ooguri and Tachikawa

The article Superconformal Algebras and String Compactification on Manifolds of SU(n) holonomy will be the basis of our group reading in Michaelmas as it contains all the ingredients needed for the calculational part of the project.

More resources will be provided if you choose this project.