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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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Torus actions whose equivariant cohomology is cohen-macaulay
Oliver Goertsches,Dirk Töben +1 more
Abstract: We study Cohen–Macaulay actions, a class of torus actions on manifolds, possibly without fixed points, which generalizes and has analogous properties as equivariantly formal actions. Their equivariant cohomology algebras are computable in the sense that a Chang–Skjelbred Lemma, and its stronger version, the exactness of an Atiyah–Bredon sequence, hold. The main difference is that the fixed-point set is replaced by the union of lowest dimensional orbits. We find sufficient conditions for the Cohen–Macaulay property such as the existence of an invariant Morse–Bott function whose critical set is the union of lowest dimensional orbits, or open-face-acyclicity of the orbit space. Specializing to the case of torus manifolds, that is, 2r-dimensional orientable compact manifolds acted on by r-dimensional tori, the latter is similar to a result of Masuda and Panov, and the converse of the result of Bredon that equivariantly formal torus manifolds are open-face-acyclic.
Journal ArticleDOI
On a generalized uncertainty principle, coherent states, and the moment map
TL;DR: In this article, it was shown that the generalized coherent states of Rawnsley minimize the uncertainty relations for any pair of generalized canonical conjugate variables (in the Kahler case) in the framework of a geometric interpretation of the Heisenberg uncertainty relations.
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Łojasiewicz–Simon gradient inequalities for analytic and Morse–Bott functions on Banach spaces
Journal ArticleDOI
Relative normal modes for nonlinear Hamiltonian systems
TL;DR: In this article, an estimate on the number of distinct relative periodic orbits around a stable relative equilibrium in a Hamiltonian system with continuous symmetry is given, which constitutes a generalization to the Hamiltonian symmetric framework of a classical result by Weinstein and Moser on the existence of periodic orbits in the energy levels surrounding a stable equilibrium.
Dissertation
Symplectic Geometry and Isomonodromic Deformations
TL;DR: In this article, the authors studied the natural symplectic geometry of moduli spaces of meromorphic connections over Riemann surfaces and showed that the moduli space of monodromy data is an infinite dimensional manifold.
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).