Steven Rayan, University of Toronto
“Hyperpolygons, Hitchin systems, and Hausel-Thaddeus mirror symmetry”
Just as the moduli space of semistable n-gons in complex projective space can be realized as the representation space of a certain shape of quiver, their hyperkaehler analogues (moduli spaces of hyper-n-gons) are quiver varieties for the same quiver, doubled. We study the cohomology of hyperpolygon spaces, and show that, at least for rank up to and including 3, they carry the structure of complete integrable systems. To accomplish this, we relate hyperpolygon spaces to singular parabolic Hitchin systems. This is joint work with Jonathan Fisher (arXiv:1410.6467). We also speculate on applications of our methods to topological mirror symmetry for moduli spaces of Higgs bundles, in the sense of Hausel-Thaddeus.