Affine Yang–Mills–Higgs metrics

Biswas, Indranil; Loftin, John; Stemmler, Matthias

doi:10.4310/jsg.2013.v11.n3.a4

Cited by 6 publications

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“…It would be interesting to explore further couplings of a projective structure to a principal connection with higher-dimensional or nonabelian structure group, or to some further data on an associated vector bundle. In the context of flat bundles on affine manifolds, something along these lines has been studied in [12].…”

Section: Introductionmentioning

confidence: 99%

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Conelike radiant structures

Fox¹

2021

Preprint

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The notion of conelike radiant structure formalizes the idea of a not necessarily flat affine connection equipped with a family of surfaces that behave like the intersections of the planes through the origin with a convex cone in a real vector space. A radiant structure means a torsion-free affine connection and a radiant vector field, meaning its covariant derivative is the identity endomorphism. A radiant structure is conelike if for every point and every two-dimensional subspace containing the radiant vector field there is a totally geodesics surface passing through the point and tangent to the subspace. Such structures exist on the total space of any principal bundle with one-dimensional fiber and on any Lie group with a quadratic structure on its Lie algebra.The affine connection of a conelike radiant structure can be normalized in a canonical way to have antisymmetric Ricci tensor. Applied to a conelike radiant structure on the total space of a principal bundle with one-dimensional fiber this yields a generalization of the classical Thomas connection of a projective structure. The compatibility of radiant and conelike structure with metrics is investigated and yields a construction of connections for which the symmetrized Ricci curvature is a constant multiple of a compatible metric that generalizes well-known constructions of Riemannian and Lorentzian Einstein-Weyl structures over Kähler-Einstein manifolds having nonzero scalar curvature. A formulation of Einstein equations for statistical manifolds (and a conformal generalization of statistical structures) is given that generalizes the Einstein-Weyl equations and encompasses these more general examples.Finally, there are constructed left-invariant conelike radiant structures on a Lie group endowed with a left-invariant nondegenerate bilinear form, and the case of three-dimensional unimodular Lie groups is described in detail.

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Section: Introductionmentioning

confidence: 99%

“…12) in which, for example, c * (ω R * )( d dt ) = c −1 ċ. Lemma 6.12. Let ρ : N → M be a principal G-bundle and let β be a principal G-connection on N , where G is R * or S 1 .…”

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confidence: 99%

Conelike radiant structures

Fox¹

2021

Preprint

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“…When the base space (X, D, g, ν) is a compact special affine manifold equipped with an affine Gauduchon metric g, Loftin ( [19]) established a Donaldson-Uhlenbeck-Yau type theorem. He proved that if a flat complex vector bundle E over (X, D, g, ν) is stable, then there is an affine Hermitian-Einstein metric on E. Biswas, Loftin and Stemmler ( [3]) also studied the flat Higgs bundle case. The definition of the flat Higgs bundle will be introduced in section 2.…”

Section: Introductionmentioning

confidence: 99%

“…rank(E)Vol(X) . We should remark that Biswas, Loftin and Stemmler ( [3]) only proved the Donaldson-Uhlenbeck-Yau type theorem in compact case. They used the continuity method and followed the argument of Lübke and Teleman ([20]).…”

Section: Introductionmentioning

confidence: 99%

Flat Higgs bundles over non-compact affine Gauduchon manifolds

Shen¹,

Zhang²,

Zhang³

2019

Preprint

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“…An affine manifold is called special if it admits a volume form which is covariant constant with respect to this flat connection D. The Donaldson-Uhlenbeck-Yau correspondence, which says that a holomorphic vector bundle V on a compact Kähler manifold admits a Yang-Mills metric if and only if V is polystable [7,11], extends to flat vector bundles on compact special affine manifolds [9]. In [2], flat Higgs vector bundles on affine manifolds were introduced and a Donaldson-Uhlenbeck-Yau type correspondence for such bundles was established. In other words, a flat Higgs vector bundle over a compact connected special…”

Section: Introductionmentioning

confidence: 99%

Approximate Yang-Mills-Higgs metrics on flat Higgs bundles over an affine manifold

Biswas

Loftin

Stemmler

2014

Communications in Analysis and Geometry

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Given a flat Higgs vector bundle (E, ∇, ϕ) over a compact connected special affine manifold, we first construct a natural filtration of E, compatible with both ∇ and ϕ, such that the successive quotients are polystable flat Higgs vector bundles. This is done by combining the Harder-Narasimhan filtration and the socle filtration that we construct. Using this filtration, we construct a smooth Hermitian metric h on E and a smooth one-parameter family {A t } t∈R of C ∞ automorphisms of E with the following property. Let ∇ t and ϕ t be the flat connection and flat Higgs field, respectively, on E constructed from ∇ and ϕ using the automorphism A t . If θ t denotes the extended connection form on E associated to the triple h, ∇ t and ϕ t , then as t −→ +∞, the connection form θ t converges in the C ∞ Fréchet topology to the extended connection form θ on E given by the affine Yang-Mills-Higgs metrics on the polystable quotients of the successive terms in the above mentioned filtration. In particular, as t −→ +∞, the curvature of θ t converges in the C ∞ Fréchet topology to the curvature of θ.

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Affine Yang–Mills–Higgs metrics

Cited by 6 publications

References 0 publications

Conelike radiant structures

Conelike radiant structures

Flat Higgs bundles over non-compact affine Gauduchon manifolds

Approximate Yang-Mills-Higgs metrics on flat Higgs bundles over an affine manifold

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