Genus-zero modular functors and intertwining operator algebras

Abstract

In [H5] (q-alg/9512024) and [H7] (q-alg/9704008), the author introduced the notion of intertwining operator algebra, a nonmeromorphic generalization of the notion of vertex operator algebra involving monodromies. The problem of constructing intertwining operator algebras from representations of suitable vertex operator algebras were solved implicitly earlier in [H3] (q-alg/9505019). In the present paper, we generalize the geometric and operadic formulation of the notion of vertex operator algebra given in [H1], [H2], [HL1] (hep-th/9301009), [HL2] and [H6] to the notion of intertwining operator algebra. We show that the category of intertwining operator algebras of central charge c is isomorphic to the category of algebras over rational genus-zero modular functors (certain analytic partial operads) of central charge c satisfying certain generalized meromorphicity. This result is one main step in the construction of genus-zero conformal field theories from representations of vertex operator algebras announced in [H5] (q-alg/9512024). One byproduct of the proof of the present isomorphism theorem is a geometric construction of (framed) braid group representations from intertwining operator algebras and thus from representations of suitable vertex operator algebras.