Abstract
We review some recent results in the theory of affine manifolds and bundles on them. Donaldson–Uhlenbeck–Yau type correspondences for flat vector bundles and principal bundles are shown. We also consider flat Higgs bundles and flat pairs on affine manifolds. A bijective correspondence between polystable flat Higgs bundles and solutions of the Yang–Mills–Higgs equation in the context of affine manifolds is shown. Also shown, in the context of affine manifolds, is a bijective correspondence between polystable flat pairs and solutions of the vortex equation.
Highlights
An affine manifold is a smooth real manifold M equipped with a flat torsion-free connection D on its tangent bundle
Our aim here is to review some recent results in the theory of flat vector bundles and flat principal bundles over a compact connected special affine manifold equipped with an affine Gauduchon metric
Let (M, D, ν) be a compact connected special affine manifold equipped with an affine Gauduchon metric g
Summary
An affine manifold is a smooth real manifold M equipped with a flat torsion-free connection D on its tangent bundle. Cheng and Yau showed that on a closed special affine manifold, an affine Kähler metric (if it exists) can be deformed to a flat metric by adding the Hessian of a smooth function; see [10]. Our aim here is to review some recent results in the theory of flat vector bundles and flat principal bundles over a compact connected special affine manifold equipped with an affine Gauduchon metric. An important aspect of the proofs of these statements is that the above correspondence between flat vector bundles over an affine manifold M and holomorphic vector bundles over the complex manifold MC ensures that local calculations can be done exactly in the same way as in the complex case. It would be interesting to try to establish these in the frame-work of affine manifolds
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