On Axiomatic Approaches to Intertwining Operator Algebras

TL;DR

This paper rigorously develops the axiomatic framework for intertwining operator algebras, proving key properties and equivalences, and constructing fundamental isomorphisms satisfying the Moore-Seiberg equations.

Contribution

It provides the first complete proofs of prior results, generalizes to non-irreducible modules, and establishes the equivalence of multiple definitions of intertwining operator algebras.

Findings

Constructed fusing and braiding isomorphisms.

Proved the Moore-Seiberg equations for genus-zero.

Established equivalences among axioms and definitions.

Abstract

We study intertwining operator algebras introduced and constructed by Huang. In the case that the intertwining operator algebras involve intertwining operators among irreducible modules for their vertex operator subalgebras, a number of results on intertwining operator algebras were given in [H9] but some of the proofs were postponed to an unpublished monograph. In this paper, we give the proofs of these results in [H9] and we formulate and prove results for general intertwining operator algebras without assuming that the modules involved are irreducible. In particular, we construct fusing and braiding isomorphisms for general intertwining operator algebras and prove that they satisfy the genus-zero Moore-Seiberg equations. We show that the Jacobi identity for intertwining operator algebras is equivalent to generalized rationality, commutativity and associativity properties of intertwining operator algebras. We introduce the locality for intertwining operator algebras and show that the Jacobi identity is equivalent to the locality, assuming that other axioms hold. Moreover, we establish that any two of the three properties, associativity, commutativity and skew-symmetry, imply the other (except that when deriving skew-symmetry from associativity and commutativity, more conditions are needed). Finally, we show that three definitions of intertwining operator algebras are equivalent.