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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
Citations
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The critical CoHA of a self dual quiver with potential
TL;DR: In this paper, it was shown that for a quiver with potential that satisfies a notion of self duality, the critical cohomological Hall algebra is supercommutative, and admits a kind of localised coproduct, which is enough to guarantee freeness.
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Banach Algebras and Rational Homotopy Theory
TL;DR: In this paper, the rational cohomology groups of unital commutative Banach algebras with maximal ideal space are determined in terms of a topological invariant associated to the last columns of the last column.
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Classical phase space singularities and quantization
TL;DR: In this paper, the authors present an overview of a research program whose aim is to develop a holomorphic quantization procedure on stratified Kaehler spaces for simple classical mechanical systems and solution spaces of classical field theories.
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Geometric Langlands And The Equations Of Nahm And Bogomolny
TL;DR: In this article, a principal SL2 subgroup of Gv makes an unexpected appearance, which can be explained using gauge theory, as this paper will show, with the help of the equations of Nahm and Bogomolny.
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Parabolic vortex equations and instantons of infinite energy
TL;DR: In this paper, the vortex equations on parabolic bundles over a Riemann surface were studied and a Hitchin-Kobayashi-type correspondence relating the existence of solutions to a certain stability condition was established.
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).