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Moduli of wild Higgs bundles on $\mathbb{C}P^1$ with $\mathbb{C}^\times$-actions

Part of: Global differential geometry

Published online by Cambridge University Press:  03 June 2021

LAURA FREDRICKSON  and
ANDREW NEITZKE
LAURA FREDRICKSON
Affiliation:
Department of Mathematics, University of Oregon, Eugene, OR97403, U.S.A. e-mail: lfredric@uoregon.edu
ANDREW NEITZKE
Affiliation:
Department of Mathematics, Yale University, Attn: Andrew Neitzke, PO Box 208283, New Haven, CT06520-8283, U.S.A. e-mail: andrew.neitzke@yale.edu
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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 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.

MSC classification

Primary: 53C07: Special connections and metrics on vector bundles (Hermite-Einstein-Yang-Mills)
Secondary: 53C56: Other complex differential geometry
Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 171 , Issue 3 , November 2021 , pp. 623 - 656
Copyright
© The Author(s), 2021. Published by Cambridge University Press on behalf of Cambridge Philosophical Society

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