We study a set $\mathcal{M}_{K,N}$ parameterising filtered SL(K)-Higgs bundles over $\mathbb{C}P^1$ with an irregular singularity at $z = \infty$, such that the eigenvalues of the Higgs field grow like $\vert \lambda \vert \sim \vert z^{N/K} \mathrm{d}z \vert$, where K and N are coprime. $\mathcal{M}_{K,N}$ carries a $\mathbb{C}^\times$-action analogous to the famous $\mathbb{C}^\times$-action introduced by Hitchin on the moduli spaces of Higgs bundles over compact curves. The construction of this $\mathbb{C}^\times$-action on $\mathcal{M}_{K,N}$ involves the rotation automorphism of the base $\mathbb{C}P^1$. We classify the fixed points of this $\mathbb{C}^\times$-action, and exhibit a curious 1-1 correspondence between these fixed points and certain representations of the vertex algebra $\mathcal{W}_K$ ; in particular we have the relation $\mu = {k-1-c_{\mathrm{eff}}}/{12}$ , where $\mu$ is a regulated version of the L2 norm of the Higgs field, and $c_{\mathrm{eff}}$ is the effective Virasoro central charge of the corresponding W-algebra representation. We also discuss a Białynicki–Birula-type decomposition of $\mathcal{M}_{K,N}$ , where the strata are labeled by isomorphism classes of the underlying filtered vector bundles.

中文翻译：

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野生希格斯粒子束的模数与 -actions

我们研究一组$\mathcal{M}_{K,N}$参数化过滤SL(ķ)-Higgs 束在$\mathbb{C}P^1$具有不规则奇点$z = \infty$，使得希格斯场的特征值增长为$\vert \lambda \vert \sim \vert z^{N/K} \mathrm{d}z \vert$， 在哪里ķ和ñ是互质的。$\mathcal{M}_{K,N}$携带一个$\mathbb{C}^\times$- 类似于著名的动作$\mathbb{C}^\times$-Hitchin 对紧曲线上希格斯束的模空间引入的作用。这座建筑的$\mathbb{C}^\times$- 采取行动$\mathcal{M}_{K,N}$涉及基的旋转自同构$\mathbb{C}P^1$. 我们对这个的不动点进行分类$\mathbb{C}^\times$-action，并在这些固定点和顶点代数的某些表示之间表现出奇怪的 1-1 对应关系$\mathcal{W}_K$; 特别是我们有关系$\mu = {k-1-c_{\mathrm{eff}}}/{12}$， 在哪里$\亩$是规范的版本大号2希格斯场范数，和$c_{\mathrm{eff}}$是相应的有效 Virasoro 中心电荷W-代数表示。我们还讨论了 Białynicki-Birula 型分解$\mathcal{M}_{K,N}$，其中地层由底层过滤矢量束的同构类标记。