Modular-Invariance of Trace Functions¶in Orbifold Theory and Generalized Moonshine

Abstract

The goal of the present paper is to provide a mathematically rigorous foundation to certain aspects of the theory of rational orbifold models in 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 number of g-twisted sectors for any automorphism g of V of finite order. We prove that the trace functions and correlation functions associated with such twisted sectors are holomorphic functions in the upper half-plane and, under suitable conditions, afford a representation 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.

Torus n-Point Functions for $${\mathbb{R}}$$ -graded Vertex Operator Superalgebras and Continuous Fermion Orbifolds

“…A number of similar results appear in [DLM1]. However, the cases k = 1, 2 are treated separately there and only for rational λ i.e.…”

“…Indeed, one can prove the analytic, elliptic and modular properties of npoint functions for many VOAs from these recursion formulae (op.cit.). This technique has since been extended to include orbifold VOAs with a g-twisted module for a finite order automorphism g [DLM1], 1 2 Z-graded VOSAs [DZ1] and Z-graded VOSAs [DZ2]. Here we consider a further generalization to obtain recursion formulae for torus n-point functions for an R-graded VOSA.…”

Symmetry breaking boundaries

“…(Let us stress that -unlike e.g. in [21] -at this point we do not require that the map σ leaves the vacuum and the Virasoro element fixed.) As already outlined in section 7, each such map is accompanied by a permutation σ * of the label set I := {λ} of A-primaries and by twisted intertwiners Θ σ ≡ Θ (λ) σ : H λ → H σ * λ between the corresponding irreducible A-modules.…”

Some irrational Generalised Moonshine from orbifolds

“…If g, h = u for some u ∈ M then (3) can always be transformed to a regular Thompson series (2) [11], [13], [8]. In particular this is always possible when o(g) and o(h) are coprime.…”