Conformal Block Divisors for Discrete Series Virasoro VOA $\text{Vir}_{2k+1,2}$

Abstract

In this work, we study a family of vector bundles on the moduli space of curves constructed from representations of $\text{Vir}_{2k+1,2}$, a family of vertex operator algebras derived from the Virasoro Lie algebra. Using the relationship between rank and degree, we characterize their asymptotic behavior, demonstrating that their first Chern classes are nef on $\overline{\rm{M}}_{g,n}$ in many cases. This is the first nontrivial example of divisors arising from vertex operator algebras that are uniformly positive for all genera. Furthermore, for $g = 1$, these divisors form a $\mathbb{Q}$-basis of the Picard group of $\overline{\rm{M}}_{1,n}$, with several desirable functorial properties. Using this basis, we characterize line bundles on certain contractions of $\overline{\rm{M}}_{1,n}$. We also propose conjectures regarding the conformal blocks of Virasoro VOAs and potential generalizations. In particular, by introducing a generalization of conformal block divisors, we provide a nonlinear nef interpolation between affine and Virasoro conformal block divisors.