Osterwalder-Schrader axioms for unitary full vertex operator algebras

TL;DR

This paper establishes Osterwalder-Schrader axioms for unitary full vertex operator algebras, connecting algebraic structures with conformal field theory and providing conditions under which correlation functions behave well.

Contribution

It introduces unitarity and spectral conditions for full VOA and proves these imply Osterwalder-Schrader axioms for correlation functions.

Findings

Correlation functions are tempered distributions.

Full VOA correlation functions satisfy conformal Osterwalder-Schrader axioms.

Heisenberg VOA extensions exemplify the theory.

Abstract

Full Vertex Operator Algebras (full VOA) are extensions of two commuting Vertex Operator Algebras, introduced to formulate compact two-dimensional conformal field theory. We define unitarity, polynomial energy bounds and polynomial spectral density for full VOA. Under these conditions and local $C_1$-cofiniteness of the simple full VOA, we show that the correlation functions of quasi-primary fields define tempered distributions and satisfy a conformal version of the Osterwalder-Schrader axioms, including the linear growth condition. As an example, we show that a family of full extensions of the Heisenberg VOA satisfies all these assumptions.