This talk will focus on geometric realizations of vertex algebras. The Virasoro uniformization provides an incarnation of the Virasoro algebra in the tangent space of the Hodge line bundle on moduli of curves with marked points and local coordinates. This allows to assign to certain representations of the Virasoro algebra a sheaf on moduli of curves together with a projective connection.

After reviewing some facts on vertex algebras and moduli of curves, I will discuss the sheaves on moduli of stable curves obtained from coinvariants of modules over conformal vertex algebras, and identify their projective connection. This is joint work with Chiara Damiolini and Angela Gibney.