The Yang-Mills heat flow on the moduli space of framed bundles on a surface

Woodward, Chris

doi:10.1353/ajm.2006.0017

Cited by 14 publications

(12 citation statements)

References 38 publications

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“…Therefore, following the method of [18] p156 (see also [24] p30 for more details), the estimates (63) and (64) show that…”

Section: An Interior Estimatementioning

confidence: 99%

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Morse theory for the space of Higgs bundles

Wilkin¹

2008

Communications in Analysis and Geometry

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The purpose of this paper is to prove the necessary analytic results to construct a Morse theory for the Yang-Mills-Higgs functional on the space of Higgs bundles over a compact Riemann surface. The main result is that the gradient flow with initial conditions (A ′′ , φ) converges to a critical point of this functional, the isomorphism class of which is given by the graded object associated to the Harder-Narasimhan-Seshadri filtration of (A ′′ , φ). In particular, the results of this paper show that the failure of hyperkähler Kirwan surjectivity for rank 2 fixed determinant Higgs bundles does not occur because of a failure of the existence of a Morse theory.

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“…Therefore, following the method of [18] p156 (see also [24] p30 for more details), the estimates (63) and (64) show that…”

Section: An Interior Estimatementioning

confidence: 99%

“…Using the identities ( 26) and ( 27) together with the equation (24) shows that the right-hand side of (30…”

Section: Existence and Uniqueness Of The Gradient Flowmentioning

confidence: 99%

Morse theory for the space of Higgs bundles

Wilkin¹

2008

Communications in Analysis and Geometry

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“…Each S j is a complex submanifold of G n , which contains C j as a deformation retract [30,18,22]. The unique open stratum S 0 coincides with the set (G n ) ss (µ n − a) of all semi-stable points in G n (Theorem 2.10).…”

Section: The Levels Of Moment Maps On Algebraic Loopsmentioning

confidence: 97%

Connectivity properties of moment maps on based loop groups

et al. 2006

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For a compact, connected, simply-connected Lie group G, the loop group LG is the infinite-dimensional Hilbert Lie group consisting of H^1-Sobolev maps S^1-->G. The geometry of LG and its homogeneous spaces is related to representation theory and has been extensively studied. The space of based loops Omega(G) is an example of a homogeneous space of $LG$ and has a natural Hamiltonian T x S^1 action, where T is the maximal torus of G. We study the moment map mu for this action, and in particular prove that its regular level sets are connected. This result is as an infinite-dimensional analogue of a theorem of Atiyah that states that the preimage of a moment map for a Hamiltonian torus action on a compact symplectic manifold is connected. In the finite-dimensional case, this connectivity result is used to prove that the image of the moment map for a compact Hamiltonian T-space is convex. Thus our theorem can also be viewed as a companion result to a theorem of Atiyah and Pressley, which states that the image mu(Omega(G)) is convex. We also show that for the energy functional E, which is the moment map for the S^1 rotation action, each non-empty preimage is connected.Comment: This is the version published by Geometry & Topology on 28 October 200

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“…For readers who may want to compare the setting here with that in the work of Teleman and Woodward [71], we briefly mention the relationship between M and the moduli of holomorphic G C -bundles. We temporarily ignore analytic issues, for which the reader should consult more detailed discussions in [27, p. 101], [76] for example. Let Σ be a closed Riemann surface with genus g. Let D be a small disk in Σ, and let Σ = Σ\D • be the Riemann surface with boundary obtained by removing the interior D • .…”

Section: And Moreover D *mentioning

confidence: 99%

“…The first type is a non-abelian localization formula (Theorem 3.19), which shows that the index is expressible as a sum of contributions localized near the components of the critical set of the norm-square of the moment map µ M : M → Lg * . There is a large literature on non-abelian localization in various forms, for example [39,65,61,72] amongst many others; in the more specific context of Hamiltonian loop group spaces references include [24,76,43,48,44]. We prove non-abelian localization by adapting a technique of Bismut-Lebeau [21, Chapter IX] to analyse the resolvents of a 1-parameter family of operators in the limit as the parameter goes to infinity.…”

Section: Introductionmentioning

confidence: 99%

Hamiltonian loop group spaces and a theorem of Teleman and Woodward

Loizides¹

2021

Preprint

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We study a natural family of elliptic boundary problems on a compact surface Σ parametrized by a non-compact moduli space of flat G-connections with framings along ∂Σ. We prove that the family has a well-defined equivariant analytic index and derive various cohomological formulas (non-abelian, delocalized, abelian). When Σ has a single boundary component and one takes the invariant part of the index, our abelian localization formula reproduces the Teleman-Woodward formula for the index of the Atiyah-Bott classes on the moduli stack of G C bundles. We carry out the analysis more generally for suitable Fredholm families over Hamiltonian loop group spaces.

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The Yang-Mills heat flow on the moduli space of framed bundles on a surface

Cited by 14 publications

References 38 publications

Morse theory for the space of Higgs bundles

Morse theory for the space of Higgs bundles

Connectivity properties of moment maps on based loop groups

Hamiltonian loop group spaces and a theorem of Teleman and Woodward

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