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.