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

We give criteria for determining the positivity of line bundles coming from vertex operator algebras (VOAs) on the moduli space ${\bar{\mathrm{M}}}_{0,n}$ 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 ${\bar{\mathrm{M}}}_{0,n}$ from certain parafermion VOAs. These give the first examples of commutant VOAs producing positive line bundles.