Relations in singular instanton homology

TL;DR

This paper computes the singular instanton homology with local coefficients for simple n-strand braids in S^1 x S^2, revealing algebraic structures linked to quantum cohomology of a Fano variety.

Contribution

It provides explicit calculations of singular instanton homology for all odd n in a specific 3-manifold, connecting it to algebraic geometry and quantum cohomology.

Findings

Homology groups described via algebraic curves

Module structures explicitly characterized

Expected equivalence with quantum cohomology ring

Abstract

We calculate the singular instanton homology with local coefficients for the simplest n-strand braids in $S^1 \times S^2$ for all odd n, describing these homology groups and their module structures in terms of the coordinate rings of explicit algebraic curves. The calculation is expected to be equivalent to computing the quantum cohomology ring of a certain Fano variety, namely a moduli space of stable parabolic bundles on a sphere with n marked points.