Modular data: the algebraic combinatorics of conformal field theory

This paper is primarily intended as an introduction for mathematicians to some of the rich algebraic combinatorics arising in for instance conformal field theory (CFT). It is essentially self-contained, apart from some of the background motivation (Section I) and examples (Section III) which are included to give the reader a sense of the context. Detailed proofs will appear elsewhere. The theory is still a work-in-progress, and emphasis is given here to several open questions and problems. In Segal’s axioms of CFT [79], any Riemann surface with boundary is assigned a certain linear homomorphism. Roughly speaking, Borcherds [12] axiomatised this data corresponding to a sphere with 3 disks removed, and the result is called a vertex operator algebra. Here we do the same with the data corresponding to a torus (and to a lesser extent a cylinder). The result is considerably simpler, as we shall see. Closely related to this, Moonshine in its more general sense involves the assignment of modular (automorphic) functions or forms to certain algebraic structures, e.g. theta functions to lattices, or vector-valued Jacobi forms to affine algebras, or Hauptmoduls to the Monster. This paper explores an important facet of Moonshine theory: the associated modular group representation. From this perspective, Monstrous Moonshine is maximally uninteresting — the corresponding representation is completely trivial! In fact, explaining that triviality was the hard part of Borcherds’ proof [13] of the Monstrous Moonshine conjectures. Let’s focus now on the former context. A rational conformal field theory∗ (RCFT) has two vertex operator algebras (VOAs) V,V ′. For simplicity we will take them to be isomorphic (otherwise the RCFT is called ‘heterotic’). The VOA V will have finitely many irreducible modules A. Consider their (normalised) characters