TASI Lectures on Moonshine

Abstract

The word moonshine refers to unexpected relations between two distinct mathematical structures: finite group representations and modular objects. It is believed that the key to understanding moonshine is through physical theories with special symmetries. Recent years have seen a variety of new ways in which finite group representations and modular objects can be connected to each other, and these developments have brought promises but also puzzles into the string theory community. These lecture notes aim to bring graduate students in theoretical physics and mathematical physics to the forefront of this active research area. In Part II of this note, we review the various cases of moonshine connections, ranging from the classical monstrous moonshine established in the last century to the most recent findings. In Part III, we discuss the relation between the moonshine connections and physics, especially string theory. After briefly reviewing a recent physical realisation of monstrous moonshine, we will describe in some details the mystery of the physical aspects of umbral moonshine, and also mention some other setups where string theory black holes can be connected to moonshine. To make the exposition self-contained, we also provide the relevant background knowledge in Part I, including sections on finite groups, modular objects, and two-dimensional conformal field theories. This part occupies half of the pages of this set of notes and can be skipped by readers who are already familiar with the relevant concepts and techniques.

Fun with F24

“…The group W aff (E 8 ) acts on the set of simple roots of W : W (E 8 ) is the subgroup that fixes a given simple root (say x 0 ), while the E 8 factor in W aff (E 8 ) acts by translations by E 8 lattice vectors. Since the multiplicities of both the odd and the even roots of g only depend on their norm, they are actually invariant under the full group of automorphisms of I 1,9 , and in particular underW .…”

“…In this paper, we will rather emphasize intricate Lie algebraic structures hidden in the deceptively simple physics of 2d free fermions. 1 As has been emphasized in several corners of the mathematical physics landscape (see, to give an incomplete list, [24,25,54] and references therein), systems with certain distinguished numbers of fermions can enjoy special properties and intertwine with several species of modular objects; our interest will be in various symmetry structures present in a system of 24 chiral fermions and some associated automorphic forms.…”

“…2 A self-dual SVOA W is one that is rational and the unique irreducible W-module (up to isomorphism). 3 For reviews of moonshine, see, e.g., [1,21].…”