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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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A spectral curve approach to Lawson symmetric CMC surfaces of genus 2
TL;DR: In this paper, a spectral curve theory for the Lawson surface of genus 2 has been proposed, where Hitchin's abelianization procedure is used to write flat line connections explicitly in terms of flat line bundles on a complex 1-dimensional torus.
XIX Simposio Internacional de Métodos Matemáticos Aplicados a las Ciencias
TL;DR: In this article, the authors present the Centro de Investigaciones en Matematicas Puras y Aplicadas (CIMPA) 2014, which is an extension of the Center of Matematics for Matemasis Puras and Aplicas (CMPA) of Costa Rica.
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The pontryagin rings of moduli spaces of arbitrary rank holomorphic bundles over a riemann surface
Richard Earl,Frances Kirwan +1 more
TL;DR: Weitsman et al. as mentioned in this paper proved that the Pontryagin ring of M(n,d) vanishes in degrees above 2n(n-1)(g-1) and that this bound is strict.
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Algebraic Weaves and Braid Varieties
TL;DR: Casals et al. as mentioned in this paper developed a diagrammatic calculus for correspondences between braid varieties and used these correspondences to obtain interesting stratifications of braid manifolds and their quotients.
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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, that any extension of semi-stable Higgs bundles 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.
Journal ArticleDOI
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).