Vertex Operators and Arithmetic: How a Single Photon Illuminates Number Theory

Summary

Introduction

This paper is based on a talk I gave at the Spitalfields Day, held under the auspices of the London Mathematical Society during the Edinburgh Conference on Moonshine. As such, it is intended for nonspecialists.

Consider the following confluence of ideas:

• As closed strings move in space-time they sweep out a worldsheet which carries the structure of a Riemann surface.

• Riemann surfaces occur as quotients Г \ H of the complex upper half-plane H by arithmetic groups Г ⊆ SL(2,R).

• The modular forms associated to Г \ H carry arithmetic information manifested in the coefficients of the corresponding Fourier series.

• Vertex operator algebra theory may be construed as an algebraicization of aspects of the theory of elementary particles (bosonic strings) and their interactions.

• Monstrous Moonshine relates certain distinguished elliptic modular functions (so-called hauptmoduln) to the Monster simple group M.

• M is the automorphism group of a particular vertex operator algebra, the Moonshine Module V^♭, which models bosons in the critical dimension c = 24 as a Z₂-orbifold.

The reader who is not conversant with all of these ideas should have no fear. The statements above are intended only as background, and to suggest several points. The first is the inevitability of connections between elliptic modular forms, vertex operator algebras and finite groups; the second is the intimate connection with ideas from theoretical physics (string theory and conformal field theory).