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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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Convexity of generalized numerical range associated with a compact lie group
TL;DR: In this paper, Westwick's convexity t heorem on the numerical range is generalized in the context of compact connected Lie groups, and the authors show how to use this generalization for compact Lie groups.
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Representations of the Kauffman bracket skein algebra III: closed surfaces and naturality
Francis Bonahon,Helen Wong +1 more
TL;DR: In this paper, it was shown that when the surface is closed, every character of the skein algebra is a classical shadow of an irreducible representation of the Kauffman bracket algebra.
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Yang–Mills theory and Tamagawa numbers: the fascination of unexpected links in mathematics
TL;DR: In this article, the authors survey the link between Yang-Mills theory and Tamagawa numbers, and explain how methods used over the last three decades to study the singular cohomology of moduli spaces of vector bundles on a smooth complex projective curve can be adapted to the setting of A^1-homotopy theory.
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A GIT interpretration of the Harder-Narasimhan filtration
TL;DR: In this paper, the authors show that the Harder-Narasimhan filtration coincides with the torsion-free sheaf fil tration of a smooth projective variety.
Journal ArticleDOI
Kirwan-Novikov inequalities on a manifold with boundary
TL;DR: In this article, the Morse-Bott inequalities for closed 1-forms with boundary were extended to manifolds with boundary, assuming only that the form is non-degenerated in the sense of Kirwan.
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).