Vertex algebras associated to modified regular representations of the Virasoro algebra

We give an abstract construction, based on the Belavin–Polyakov–Zamolodchikov equations, of a family of vertex algebras with conformal elements of rank 26 associated to the modified regular representations of the Virasoro algebra. The vertex operators are obtained from the products of intertwining operators for a pair of Virasoro algebras. We explicitly determine the structure coefficients that yield the axioms of vertex algebras. In the process of our construction, we obtain new hypergeometric identities.