Osterwalder-Schrader axioms for unitary full vertex operator algebras

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.