Vertex Algebras Associated to Integral Lattices and Vector Bundles on the Projective Line

There is a general construction producing a vector bundle on any smooth algebraic curve from a finite-dimensional representation of the "positive half" $\textnormal{Vir}_+$ of the Virasoro algebra. We study this construction in detail when the curve is the projective line. We prove, in particular, that in this case the bundle is completely determined by the action of the operators $L_0$ and $L_1$ and show how to compute it explicitly (i.e., to find the decomposition into the direct sum of line bundles) knowing the Jordan decomposition of $L_1$. After that, we compute explicitly these bundles for two families of representations of $\textnormal{Vir}_+$: the Fock modules and the Demazure modules of level $1$ corresponding to a multiple of the first fundamental weight for the affine Kac-Moody algebra $\widehat{\mathfrak{sl}}(n,\CC)$