Unitary vertex algebras and Wightman conformal field theories

TL;DR

This paper establishes a deep equivalence between unitary M"obius vertex algebras and Wightman conformal field theories on the circle, providing new insights into their analytic and operator-theoretic structures.

Contribution

It proves an equivalence between unitary vertex algebras and Wightman conformal field theories, linking algebraic and analytic frameworks in conformal field theory.

Findings

New analytic and operator-theoretic insights into vertex operators

Characterization of OPEs satisfying vertex algebra axioms

Connections established between vertex operator algebras and conformal nets

Abstract

We prove an equivalence between the following notions: (i) unitary M\"obius vertex algebras, and (ii) Wightman conformal field theories on the circle (with finite-dimensional conformal weight spaces) satisfying an additional condition that we call uniformly bounded order. Reading this equivalence in one direction, we obtain new analytic and operator-theoretic information about vertex operators. In the other direction we characterize OPEs of Wightman fields and show they satisfy the axioms of a vertex algebra. As an application we establish new results linking unitary vertex operator algebras with conformal nets.