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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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Journal ArticleDOI
A Direct Existence Proof for the Vortex Equations Over a Compact Riemann Surface
TL;DR: In this article, a direct existence proof for the vortex equations over a compact Riemann surface is given, exploiting the interpretation of these equations in terms of moment maps, which correspond to the absolute minima of the Yang-Mills-Higgs functional.
Journal ArticleDOI
Spectral networks
TL;DR: Spectral networks as discussed by the authors are networks of trajectories on Riemann surfaces obeying certain local rules, which arise naturally in four-dimensional N = 2 theories coupled to surface defects, particularly the theories of class S. In these theories spectral networks provide a useful tool for the computation of BPS degeneracies.
Book ChapterDOI
Morse-Bott theory and equivariant cohomology
D. M. Austin,Peter J. Braam +1 more
TL;DR: In Morse theory, the topology of a manifold is investigated in terms of these notions with equally profound success: Smale proved the h-cobordism and generalized Poincare conjectures using surgery cobordisms as discussed by the authors.
Journal ArticleDOI
Surface group representations and U(p, q)-Higgs bundles
Steven B. Bradlow,Oscar García-Prada,Oscar García-Prada,Oscar García-Prada,Peter B. Gothen,Peter B. Gothen,Peter B. Gothen +6 more
TL;DR: In this paper, the moduli spaces of U(p, q)-Higgs bundles over a Riemann surface were studied using the L2 norm of the Higgs field as a Morse function.
Posted Content
Differential Geometry of Gerbes
Lawrence Breen,William Messing +1 more
TL;DR: In this paper, the notion of a connective structure for a gerbe on a space X is defined in a global manner and a global definition of the 3-curvature of such connective structures as a 3-form on X with values in the Lie stack of the gauge stack of a Gerbe is given.
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).