Abstract
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 L
2
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.
METRICS AND SPECIAL KÄHLER GEOMETRY ON THE MODULI SPACES OF HIGGS BUNDLES AND HITCHIN SYSTEMS
