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The Yang-Mills equations over Riemann surfaces
Michael Atiyah,Raoul Bott +1 more
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In this article, the Yang-Mills functional over a Riemann surface is studied from the point of view of Morse theory, and the main result is that this is a perfect 9 functional provided due account is taken of its gauge symmetry.Abstract:
The Yang-Mills functional over a Riemann surface is studied from the point of view of Morse theory. The main result is that this is a ‘perfect9 functional provided due account is taken of its gauge symmetry. This enables topological conclusions to be drawn about the critical sets and leads eventually to information about the moduli space of algebraic bundles over the Riemann surface. This in turn depends on the interplay between the holomorphic and unitary structures, which is analysed in detail.read more
Citations
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Yang-Mills theory over surfaces and the Atiyah-Segal theorem
TL;DR: In this paper, Morse theory for the Yang-Mills functional can be used to prove an analogue, for surface groups, of the Atiyah-Segal theorem, and the main theorem provides an isomorphism in homotopy for all compact, aspherical surfaces.
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Intermediate Jacobians and Hodge structures of moduli spaces
Donu Arapura,Pramathanath Sastry +1 more
TL;DR: The mixed Hodge structure on the low degree cohomology of the moduli space of vector bundles on a curve is studied in this article, where the authors also give a proof of a Torelli theorem.
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Chern-Simons Gauge Theory and projectively flat vector bundles on 421-1421-1421-1
TL;DR: In this article, a vector bundle on Teichmuller space is considered and a natural connection on it is defined, and the curvature of the connection is shown to be projectively flat.
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The noncommutative geometry of Yang–Mills fields
TL;DR: In this article, the authors generalize the noncommutative description of Yang-Mills theory to topologically non-trivial gauge configurations and show that this generalization can be applied to topological non-Trivial configurations.
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On the blow-up set of the Yang-Mills flow on Kähler surfaces
TL;DR: The Yang-Mills flow on a Kahler surface with holomorphic initial data converges smoothly away from a singular set determined by the Harder-Narasimhan-Seshadri filtration of the initial holomorphic bundle as mentioned in this paper.
References
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Book
Principles of Algebraic Geometry
Phillip Griffiths,Joe Harris +1 more
TL;DR: In this paper, a comprehensive, self-contained treatment of complex manifold theory is presented, focusing on results applicable to projective varieties, and including discussion of the theory of Riemann surfaces and algebraic curves, algebraic surfaces and the quadric line complex.
Book
Inequalities: Theory of Majorization and Its Applications
TL;DR: In this paper, Doubly Stochastic Matrices and Schur-Convex Functions are used to represent matrix functions in the context of matrix factorizations, compounds, direct products and M-matrices.
Book
Geometric Invariant Theory
TL;DR: Geometric invariant theory for moduli spaces has been studied extensively in the mathematical community as mentioned in this paper, with a large number of applications to the moduli space construction problem, see, for instance, the work of Mumford and Fogarty.
Journal ArticleDOI
Self-duality in four-dimensional Riemannian geometry
TL;DR: In this article, the authors present a self-contained account of the ideas of R. Penrose connecting four-dimensional Riemannian geometry with three-dimensional complex analysis, and apply this to the self-dual Yang-Mills equations in Euclidean 4-space and compute the number of moduli for any compact gauge group.
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Stable and unitary vector bundles on a compact Riemann surface
M. S. Narasimhan,C. S. Seshadri +1 more
TL;DR: In this article, it was shown that on the space of isomorphic classes of unitary vector bundles on X of a given rank, there is a natural structure of a normal projective variety (Theorem 8.1).