From $S^1$-fixed points to $\mathcal{W}$-algebra representations

Abstract

We study a set ${\mathcal{M}}_{K,N}$ parameterizing filtered $SL(K)$-Higgs bundles over ${\mathbb{CP}}^1$ with an irregular singularity at $z=\infty$, such that the eigenvalues of the Higgs field grow like $|\hat{I}»|\sim |z^{N/K}\mathrm{d}z|$, 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 ${\mathbb{CP}}^1$. 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 $\hat{I}¼=\frac{1}{12}\left(K-1-c_{\mathrm{eff}}\right)$, where $\hat{I}¼$ 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 Bialynicki-Birula-type stratification of ${\mathcal{M}}_{K,N}$, where the strata are labeled by isomorphism classes of the underlying filtered vector bundles.

36 pages