Recursive relations in the cohomology rings of moduli spaces of rank 2 parabolic bundles on the Riemann sphere

TL;DR

This paper derives recursive relations in the cohomology rings of moduli spaces of rank 2 parabolic bundles on the Riemann sphere, providing formulas for their Poincaré polynomials and cohomological relations.

Contribution

It introduces recursive cohomology relations for rank 2 parabolic bundles with specific weights, extending known recursive structures from non-parabolic cases.

Findings

Derived a formula for the Poincaré polynomial of the moduli space.

Established recursive relations in the cohomology ring for genus zero case.

Connected the relations to orthogonal polynomials and continued fractions.

Abstract

We study the singular cohomology of the moduli space of rank 2 parabolic bundles on a Riemann surface where the weights are all 1/4. We give a formula, based on work of Boden, for the Poincar\'e polynomial of this moduli space in general, and deduce a complete set of relations in the cohomology ring for the case when the genus is zero. These relations are recursive in the number of parabolic points (which must be odd), and are analogous to the relations appearing in the cohomology ring for the non-parabolic story, which are recursive in the genus. Our methods use techniques from a paper of Weitsman to reduce to a linear algebra problem, which we then solve using the theory of orthogonal polynomials and a continued fraction expansion for the generating function of the Euler numbers.