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