We study the possible dimension vectors of indecomposable parabolic bundles on the projective line, and use our answer to solve the problem of characterizing those collections of conjugacy classes of n×n matrices for which one can find matrices in their closures whose product is equal to the identity matrix. Both answers depend on the root system of a Kac-Moody Lie algebra. Our proofs use Ringel’s theory of tubular algebras, work of Mihai on the existence of logarithmic connections, the Riemann-Hilbert...

We use the methods that were developed by Adler and van Moerbeke to determine explicit equations for a certain moduli space, that was studied by Narasimhan and Ramanan. Stated briefly it is, for a fixed non-hyperelliptic Riemann surface $\Gamma$ of genus $3$, the moduli space of semi-stable rank two bundles with trivial determinant on $\Gamma$. They showed that it can be realized as a projective variety, more precisely as a quartic hypersurface of ${\mathbb{P}}^7$, whose singular locus is the Kummer variety of $\Gamma$. We first construct...