Positivity of coinvariant divisors on $\overline{\mathrm{M}}_{0,n}$ and the parafermions

TL;DR

This paper establishes criteria for the positivity of line bundles derived from vertex operator algebras on the moduli space of rational curves, and constructs new positive line bundles from parafermion VOAs, highlighting novel connections between VOAs and algebraic geometry.

Contribution

It provides the first criteria for VOA-derived line bundle positivity on n and constructs new positive line bundles from parafermion VOAs, expanding the understanding of VOA applications.

Findings

Criteria for VOA line bundle positivity on n established

Constructed positive line bundles from parafermion VOAs

First examples of commutant VOAs producing positive line bundles

Abstract

We give criteria for determining the positivity of line bundles coming from vertex operator algebras (VOAs) on the moduli space $\overline{\mathrm{M}}_{0,n}$ of rational curves with $n$ marked points. The criteria use the multiplicative structure of VOA representations encoded in the fusion ring. Using them, we construct positive line bundles on $\overline{\mathrm{M}}_{0,n}$ from certain parafermion VOAs. These give the first examples of commutant VOAs producing positive line bundles.