Modular invariance of trace functions in orbifold theory
Chongying Dong, Haisheng Li, Geoffrey Mason
TL;DR
This work provides a rigorous mathematical foundation for orbifold conformal field theory of rational vertex operator algebras under finite-order automorphisms. By developing an equivariant Zhu framework with A_g(V), introducing (g,h)-torus 1-point functions and their differential equations, and proving holomorphy, modular-invariance, and rationality results, it establishes finiteness and structure for twisted sectors and their trace functions. The main contributions include a basis description of (g,h)-conformal blocks, rationality of the central charge c and conformal weights, and verification of Condition C2 for principal VOAs; together with generalized Moonshine results via Hauptmodul properties. In particular, the Moonshine module exhibits Hauptmodul behavior for cyclic pairs (g,h) and a unique twisted sector, strengthening the link between VOAs, modular forms, and finite group representations.
Abstract
The goal of the present paper is to provide a mathematically rigorous foundation to certain aspects of rational orbifold conformal field theory, in other words the theory of rational vertex operator algebras and their automorphisms. Under a certain finiteness condition on a rational vertex operator algebra V which holds in all known examples, we determine the precise numbers of g-twisted sectors for any automorphism g of V of finite order. We prove that the trace functions and correlations functions associated with such twisted sectors are holomorphic functions in the upper half-plane and, under suitable conditions, afford a representations of the modular group of the type prescribed in string theory. We establish the rationality of conformal weights and central charge. In addition to conformal field theory itself, where our conclusions are required on physical grounds, there are applications to the generalized Moonshine conjectures of Conway-Norton-Queen and to equivariant elliptic cohomology.