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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
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Smooth structures on certain moduli spaces for bundles on a surface
TL;DR: In this article, the authors show that the assignment to a smooth connection with reference to suitable closed paths yields a diffeomorphism from the moduli space of central Yang-Mills connections onto the space of representations of the universal central extension of the fundamental group of a closed surface.
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Uniqueness of extreme horizons in Einstein-Yang-Mills theory
Carmen Li,James Lucietti +1 more
TL;DR: In this article, the authors consider stationary extreme black hole solutions to the Einstein-Yang-Mills equations in four dimensions, allowing for a negative cosmological constant, and prove that any axisymmetric black hole of this kind possesses a near-horizon AdS(2) symmetry.
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Local monotonicity for the Yang–Mills–Higgs flow
TL;DR: A local monotonicity formula for the Yang-Mills-Higgs flow on G-bundles was proved in this paper, where it was shown that the monotone quantity coincides on certain self-similar solutions with that appearing in existing non-local monotonic formulae for the YMM and YMM flows.
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Degeneracy of 2-Forms and 3-Forms
TL;DR: In this article, the degree of the convergence of certain orbits of the representation of a manifold with respect to a complex 2-form and 3-form complex manifold is calculated. But the degree is not directly proportional to the number of vertices in the manifold.
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The Kauffman skein algebra of a surface at $${\sqrt{-1}}$$
TL;DR: In this article, the structure of the Kauffman algebra of a surface with parameter equal to (1 − 1) is studied and an interpretation of this algebra as an algebra of parallel transport operators acting on sections of a line bundle over the moduli space of flat SU(2)-connections over the surface is obtained.
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.
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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.
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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).