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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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Moduli spaces of bundles over Riemann surfaces and the Yang-Mills stratification revisited
TL;DR: In this article, a complete set of relations between the standard generators for the cohomology of the moduli spaces of stable holomorphic bundles of rank n and degree d when n and d are coprime and n>2 was found.
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The use of geometric and quantum group techniques for wild quivers
TL;DR: In this article, several results relating the representation theory of wild quivers to algebraic geometry and quantum group theory are discussed, as well as potential applications to the study of quivers are discussed.
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Topology of U(2, 1) Representation Spaces
TL;DR: The Betti numbers of moduli spaces of representations of a universal central extension of a surface group in the groups U(2,1) and SU(2.1) are calculated in this paper.
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Factorization of rank two theta functions. II. Proof of the Verlinde formula.
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Approximate Hermitian-Yang-Mills structures and semistability for Higgs bundles. II: Higgs sheaves and admissible structures
TL;DR: In this paper, the basic properties of Higgs sheaves over compact Kahler manifolds were studied and some results concerning the notion of semistability were established; in particular, any extension of semi-stable Higgs heaves with equal slopes is semistable.
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).