Vertex operator algebras associated to modified regular representations of

We study a family of vertex operator algebras which admit two copies of the
affine Lie algebra actions at dual levels. We will discuss two constructions:
one is to use intertwining operators and Knizhnik-Zamolodchikov equations; the
other is to use the enveloping algebra of the vertex algebroid associated to
the Lie group and a fixed level. We will show that the two constructions yield
the same vertex operator algebra.