On Certain Automorphic Forms Associated to Rational Vertex Operator Algebras

Summary

Abstract

To every rational vertex operator algebra V we associate an automorphic form on Г′ (1) that we call the Wronskian of V. We have previously shown [M2], [M3] that in the case of Virasoro minimal models it is possible to give qualitative arguments about the Wronskian by using the representation theoretic methods. Here we apply the theory of automorphic forms and extend our previous work to a larger class of vertex operator algebras. We also give a detailed analysis of two-dimensional modular invariant spaces that arise from affine Kac-Moody Lie algebras.

As a main byproduct of our analysis we provide new proofs of certain Dyson-Macdonald's identities for powers of the Dedekind η–function for C_l, BC_l and D_l series, and related identities (e.g., Jacobi's Four Square Theorem).

Introduction and notation

The existence of a fusion ring and modular invariance of graded dimensions, or characters, are the most interesting features of every rational conformal field theory [MS]. When it comes to vertex operator algebra theory, proving modular invariance [Zh] (cf. [DLM1]) and ultimately the Verlinde formula [Hu1] is a formidable task. A key ingredient in proving modular invariance is played by the so-called C₂–cofiniteness [Zh] which, in particular, guarantees the convergence of all one-point functions on the torus. The C₂–cofiniteness plays an important role in the proof of the Verlinde conjecture as well [Hu1]. Even though the vector space spanned by irreducible characters is a PSL(2,ℤ)–module (i.e., a modular invariant space), one cannot state the Verlinde formula without having an action of SL(2,ℤ) (the operator S² does not act as the identity in general–charge conjugation).