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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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Connected components of spaces of surface group representations II
Nan-Kuo Ho,Chiu-Chu Melissa Liu +1 more
TL;DR: In this paper, the connected components of the space of surface group representations for any compact connected semisimple Lie group and any closed compact (orientable or nonorientable) surface were discussed.
Proceedings ArticleDOI
Loop groups, elliptic singularities and principal bundles over elliptic curves
Stefan Helmke,Peter Slodowy +1 more
TL;DR: The relation between simple algebraic groups and simple singularities was studied in this article, where the simple singularity appeared as the generic singularity in codimension two of the unipotent variety of simple algebra groups.
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Splitting of gauge groups
Daisuke Kishimoto,Akira Kono +1 more
TL;DR: In this article, the authors studied the splitting of the exact sequence of topological groups in the category of A n -spaces and A n-maps in connection with the triviality of the adjoint bundle of P and with the higher homotopy commutativity of G.
Journal ArticleDOI
On Unstable Principal Bundles over Elliptic Curves
Stefan Helmke,Peter Slodowy +1 more
TL;DR: In this article, a relation between simple elliptic singularities in the sense of Saito [22] and certain holomorphic Kac-Moody loop groups is established which generalizes a well known theorem of Brieskorn [6], cf. also [26], relating simple singularities of type A, D, E and the corresponding simple algebraic groups G.
Posted Content
Canonical Generators for the Cohomology of Moduli of Parabolic Bundles on Curves
I. Biswas,N. Raghavendra +1 more
TL;DR: In this paper, the authors determine generators of the rational cohomology algebras of moduli spaces of parabolic vector bundles on a curve, under some primality conditions on the parabolic datum.
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).