Looijenga line bundles in complex analytic elliptic cohomology

TL;DR

This paper explores the connection between moduli spaces of elliptic curves, $U(1)$-bundles, and Looijenga line bundles, providing calculations that link these geometric objects to complex analytic elliptic cohomology.

Contribution

It introduces a novel calculation relating moduli of elliptic curves and line bundles to Borel cohomology, advancing the understanding of complex analytic equivariant elliptic cohomology.

Findings

Moduli of elliptic curves arise from Borel cohomology of $U(1)$-bundles.

Looijenga line bundles are connected to moduli spaces of principal bundles.

Speculations on the relation to complex analytic equivariant elliptic cohomology.

Abstract

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