Journal ArticleDOI
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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Harder-Narasimhan reduction of a principal bundle
Indranil Biswas,Yogish I. Holla +1 more
TL;DR: In this paper, the uniqueness of canonical reduction is proved under the assumption that the characteristic of $k$ is zero and under a mild assumption on the characteristic, the uniqueness is also proved when the characteristic is positive.
Book ChapterDOI
Global Differential Geometry
TL;DR: The only place to start a survey like this is in the present, a present which, in the words of Philip Larkin, is the future furthest childhood saw.
Journal ArticleDOI
Dirac geometry, quasi-Poisson actions and D/G-valued moment maps
Henrique Bursztyn,Marius Crainic +1 more
TL;DR: In this article, a Dirac geometric approach to Hamiltonian spaces with D/G-valued moment maps, originally introduced by Alekseev and Kosmann-Schwarzbach [3] in terms of quasi-Poisson structures, is presented.
Journal ArticleDOI
M5 brane and four dimensional \mathcal{N} = 1 theories I
TL;DR: In this paper, a generalized Hitchin's equation involving two Higgs fields is proposed as the BPS equation for regular puncture compactification, and the puncture is interpreted as the singular boundary condition of this equation.
Journal ArticleDOI
Topological methods in moduli theory
TL;DR: In this paper, the moduli spaces of algebraic curves with a group of automorphisms of a given topological type are studied in detail, following new results by the author, Michael Lonne and Fabio Perroni.
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.
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).