Looijenga line bundles in complex analytic elliptic cohomology

Abstract

We present a calculation that shows how the moduli of complex analytic elliptic curves arises naturally from the Borel cohomology of an extended moduli space of $U\left(1\right)$-bundles on a torus. Furthermore, we show how the analogous calculation, applied to a moduli space of principal bundles for a $K\left(\mathbb{Z},2\right)$ central extension of $U{\left(1\right)}^d$, gives rise to Looijenga line bundles. We then speculate on the relation of these calculations to the construction of complex analytic equivariant elliptic cohomology.