Vertex algebras are algebraic structures coming from two dimensional  
conformal field theory. This talk is about their relation with moduli  
spaces of Riemann surfaces.

I will first review some background material. In particular, I will  
recall that a vertex algebra is a graded vector space V with  
additional structures, and these structures force the Hilbert-Poincaré  
series of V, conveniently normalized, to be a modular form.

I will then associate to any holomorphic vertex algebra a collection  
of Teichmüller modular forms (= sections of powers of the lambda class  
on the moduli space of Riemann surfaces), whose expansion near the  
boundary gives back some information about the correlation functions  
of the vertex algebra. This is a generalization of the  
Hilbert-Poincaré series of V, it uses moduli spaces of Riemann  
surfaces of arbitrarily high genus, and it is sometime called  
partition function of the vertex algebra. I will also explain some  
partial results towards the reconstruction of the vertex algebra out  
of these Teichmüller modular forms.

Using the above mentioned construction, we can use vertex algebras to  
study problems about the moduli space of Riemann surfaces, such as the  
Schottky problem, the computation of the slope of the effective cone,  
and the computation of the dimension of the space of sections of  
powers of the lambda class. On the other hand, this construction  
allows us to use the geometry of the moduli space of Riemann surfaces  
to classify vertex algebras; in particular, I will discuss how  
conjectures and known results about the slope of the effective cone  
can be used to study the unicity of the moonshine vertex algebras.

This is a work in progress with Sebastiano Carpi.