扭曲的阿盖尔箭和希格斯束
Twisted argyle quivers and Higgs bundles
摘要：
通常，在向量空间范畴中，模簇被构造为模表示的模空间。在一个更丰富的范畴中考虑向量表示也是很自然的，即在某个复杂簇上的向量丛的表示，这个复杂簇具有扭曲态射的固定层。在这个扭曲的范畴中，A型箭图的表示-在文献中称为“全纯链”-在希格斯束的模空间的拓扑问题中有实际用途。在该问题中，簇是亏格至少为2的黎曼曲面，扭是它的标准线丛。我们延长治疗扭曲的A-型的Higgs表示的任何属使用希钦斯束诱导的稳定性条件和计算它们的变形理论。然后，我们特别关注所谓的“argyle quivers”，其中排名标签在1和整数$r_i\geq1$之间交替。我们给出了明确的几何标识的阿盖尔箭图的扭曲表示的模空间的$\mathbb{P}^1$使用不变理论的非约化作用，通过多项式的欧几里得约化。这导致了分层的模空间的改变，我们确定与“碰撞流形”不变的零多项式的束类型。我们还涉及到目前的工作Bradlow-Daskalopoulos稳定性和Thaddeus的拉回映射到稳定的元组。我们将所得结果应用于计算$\mathbb{P}^1$上低秩扭曲Higgs丛模空间的贝蒂数，其中Higgs场取值于任意丰富的线丛.我们的结果与由ADHM递推公式导出的广义Poincaré级数一致。
英文摘要：
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 Poincaré series arising from the ADHM recursion formula.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
