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.