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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Book ChapterDOI
Cohomology of a Moduli Space of Vector Bundles
V. Balaji,C. S. Seshadri +1 more
TL;DR: In this article, the moduli space of semi-stable vector bundles V of rank two and trivial determinant is defined over the field of complex numbers, where V is a smooth projective curve of genus g.
Journal ArticleDOI
Representations of the fundamental group of an l-punctured sphere generated by products of Lagrangian involutions
TL;DR: In this article, it was shown that the fixed-point set of an anti-symplectic involution defined on the mod- uli space is a Lagrangian submanifold of the symplectic structure of MC.
Journal ArticleDOI
Norm-square localization for Hamiltonian LG-spaces
TL;DR: In this article, a formula for twisted Duistermaat-heckman distributions associated to a Hamiltonian L G -space is presented, and the terms of the formula are localized at the critical points of the norm-square of the moment map, and can be computed in cross-sections.
Journal ArticleDOI
Differential graded Lie algebras and singularities of level sets of momentum mappings
TL;DR: In this paper, the authors discuss related ideas in a more algebraic context by associating to an affine Hamiltonian action a differential graded Lie algebra, which in the presence of an invariant positive complex structure, is formal in the sence of [5].
Dissertation
Infinite-Dimensional Lie Theory for Gauge Groups
TL;DR: In this paper, the authors consider symmetry groups of principal bundles and initiate a Lie theoretic treatment of these groups, referred to as gauge groups, modelled on an appropriate vector space.
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).