Wild non-abelian Hodge theory on curves

By O. Biquard, P. Boalch. Compositio Mathematica, 2001

237 citations recorded by the source corpus.

On a complex curve, we establish a correspondence between integrable connections with irregular singularities, and Higgs bundles such that the Higgs field is meromorphic with poles of any order. Moduli spaces of these objects are obtained with fixed generic polar parts at each singularity, which amounts to fixing a coadjoint orbit of the group $GL_r(\mathbb{C}[z]/z^n)$. We prove that they carry complete hyper-Kähler metrics.

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/2a2657214e952a151d7291465ffe15021010dfa0

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In Geometry and Topology

1. E. Witten
2. Dan Xie
3. Sergei Gukov
4. T. Mochizuki
5. P. Boalch
6. David Dumas
7. Michael Wolf
8. G. Bonelli

In Mathematical Physics

1. Dan Xie
2. E. Witten
3. Sergei Gukov
4. K. Maruyoshi
5. T. Mochizuki
6. A. Tanzini
7. G. Bonelli
8. P. Boalch

Citations in the corpus

• Quasi-Hamiltonian geometry of meromorphic connections (2002, unknown): “(In general—nongeneric exponents—the quotient will not be well behaved and stability conditions are needed, see [ 8 ].) Note that the full topological type is not necessarily fixed in M(A), although the degree (=‘deg’ below) is.”
• Polynomial cubic differentials and convex polygons in the projective plane (2014, methodology): “The self-duality equations have been studied in the irregular case by Biquard-Boalch [BB04], extending Simpson’s work for the case of a simple pole [Sim90].”
• General Argyres-Douglas theory (2012, background): “it is also a hyperkahler manifold [10].”
• Riemann–Hilbert for tame complex parahoric connections (2010, background): “Our motivation was to understand the spaces of monodromy-type data that occur in the extension of the nonabelian Hodge correspondence to the case of irregular connections on curves [23], [5], and its extension to arbitrary G.”
• Gauge Theory, Ramification, And The Geometric Langlands Program (2006, methodology): “A corresponding theory for Higgs bundles with poles of higher order has been developed by Biquard and Boalch [25].”
• Asymptotic behaviour of certain families of harmonic bundles on Riemann surfaces (2015, methodology): “We need a family of harmonic metrics given on a neighbourhood of the discriminant locus, for which we apply the Kobayashi-Hitchin correspondence for wild harmonic bundles on curves given in [2] (see [20] for the tame case).”
• Wild quiver gauge theories (2011, background): “However, we can allow a higher order singularity of the Higgs field, which is called wild ramification [28].”
• Geometry and braiding of Stokes data; Fission and wild character varieties (2011, methodology): “(Specifically, the cases here correspond to setting Re(λi) = 0 in [10] so that stability is equivalent to irreducibility.”
• Wild Harmonic Bundles and Wild Pure Twistor D-modules (2008, methodology): “The following proposition was observed by Biquard-Boalch ([9]), at least if the wild harmonic bundle is given on a quasiprojective curve.”