Twisted argyle quivers and Higgs bundles

Abstract

Ordinarily, quiver varieties are constructed as moduli spaces of quiver representations in the category of vector spaces. It is also natural to consider quiver representations in a richer category, namely that of vector bundles on some complex variety equipped with a fixed sheaf that twists the morphisms. Representations of A-type quivers in this twisted category --- known in the literature as "holomorphic chains" --- have practical use in questions concerning the topology of the moduli space of Higgs bundles. In that problem, the variety is a Riemann surface of genus at least 2, and the twist is its canonical line bundle. We extend the treatment of twisted A-type quiver representations to any genus using the Hitchin stability condition induced by Higgs bundles and computing their deformation theory. We then focus in particular on so-called "argyle quivers", where the rank labelling alternates between 1 and integers $r_i\geq1$. We give explicit geometric identifications of moduli spaces of twisted representations of argyle quivers on $\mathbb{P}^1$ using invariant theory for a non-reductive action via Euclidean reduction on polynomials. This leads to a stratification of the moduli space by change of bundle type, which we identify with "collision manifolds" of invariant zeroes of polynomials. We also relate the present work to Bradlow-Daskalopoulos stability and Thaddeus' pullback maps to stable tuples. We apply our results to computing $\mathbb{Q}$-Betti numbers of low-rank twisted Higgs bundle moduli spaces on $\mathbb{P}^1$, where the Higgs fields take values in an arbitrary ample line bundle. Our results agree with conjectural Poincar\'e series arising from the ADHM recursion formula.