Introduction to Nonabelian Hodge Theory

Garcia-Raboso, Alberto; Rayan, Steven

doi:10.1007/978-1-4939-2830-9_5

Cited by 8 publications

(4 citation statements)

References 67 publications

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“… ** For further details concerning the Riemann–Hilbert correspondence, the Narasimhan–Seshadri correspondence, stability, differentiable versus complex structures on moduli spaces, and the larger nonabelian Hodge theory correspondence that these ideas fit into, we refer the reader to the survey in ref. 64 , for instance. …”

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confidence: 99%

Automorphic Bloch theorems for hyperbolic lattices

Maciejko

Rayan

2022

Proc. Natl. Acad. Sci. U.S.A.

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Hyperbolic lattices are a new form of synthetic quantum matter in which particles effectively hop on a discrete tessellation of two-dimensional (2D) hyperbolic space, a non-Euclidean space of uniform negative curvature. To describe the single-particle eigenstates and eigenenergies for hopping on such a lattice, a hyperbolic generalization of band theory was previously constructed, based on ideas from algebraic geometry. In this hyperbolic band theory, eigenstates are automorphic functions, and the Brillouin zone is a higher-dimensional torus, the Jacobian of the compactified unit cell understood as a higher-genus Riemann surface. Three important questions were left unanswered: whether a band theory can be expected to hold for a non-Euclidean lattice, where translations do not generally commute; whether a formal Bloch theorem can be rigorously established; and whether hyperbolic band theory can describe finite lattices realized in an experiment. In the present work, we address all three questions simultaneously. By formulating periodic boundary conditions for finite but arbitrarily large lattices, we show that a generalized Bloch theorem can be rigorously proved but may or may not involve higher-dimensional irreducible representations (irreps) of the nonabelian translation group, depending on the lattice geometry. Higher-dimensional irreps correspond to points in a moduli space of higher-rank stable holomorphic vector bundles, which further generalizes the notion of Brillouin zone beyond the Jacobian. For a large class of finite lattices, only 1D irreps appear, and the hyperbolic band theory previously developed becomes exact.

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confidence: 99%

Automorphic Bloch theorems for hyperbolic lattices

Maciejko

Rayan

2022

Proc. Natl. Acad. Sci. U.S.A.

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“…We treat some properties of certain G m -fixed points on M ss Higgs (G, C). A helpful discussion of the G m -action on the moduli space can be found in [GR15].…”

Section: Moduli Of Semistable Higgs Bundles and Fixed Pointsmentioning

confidence: 99%

On the Kirwan map for moduli of Higgs bundles

Cliff¹,

Nevins²,

Shen³

2021

Alg. Geom.

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Let C be a smooth complex projective curve and G a connected complex reductive group. We prove that if the center Z(G) of G is disconnected, then the Kirwan mapHiggs (G, C), Q , from the cohomology of the moduli stack of all G-bundles to the moduli stack of semistable G-Higgs bundles, cannot be surjective. We establish similar results for intersection cohomology and for the cohomology of the coarse moduli spaces. The proof uses a Borel-Quillen-style localization result for equivariant cohomology of stacks to reduce to an explicit calculation.

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“…For a deeper discussion of the gauge theory, including an exploration of recent results concerning the global properties of the hyperkähler metric, we refer the reader to [21] in the same collection of mini-course articles -as well as of course Hitchin's original article [38]. Regarding nonabelian Hodge theory in particular, we refer the reader to works of Simpson [61,62] and to recent surveys such as [23,65].…”

Section: Gauge Theorymentioning

confidence: 99%