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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
Limiting configurations for solutions of Hitchin’s equation
TL;DR: In this article, the degeneration behavior near the ends of the moduli space of Hitchin's self-duality equation is studied in a set of generic directions by showing how limiting configurations can be desingularized.
Posted Content
Global existence and convergence of smooth solutions to Yang-Mills gradient flow over compact four-manifolds
TL;DR: In this article, the authors developed results on global existence and convergence of solutions to the gradient flow equation for the Yang-Mills energy functional on a principal bundle, with compact Lie structure group, over a closed, four-dimensional, Riemannian, smooth manifold.
Book ChapterDOI
Cauchy-Riemann Operators, Self—Duality, and the Spectral Flow
TL;DR: In this article, it was shown that the spectral flow of the self-adjoint operator family A(t) has invertible limits, where t is the dimension of the space of gradient flow lines connecting two critical points.
Journal ArticleDOI
Algebraic geometric aspects of smooth structure. I. The Donaldson polynomials
TL;DR: In this article, the Donaldson polynomials and smooth invariance of the canonical class are discussed and connections between vector bundles and metrics over 4-manifolds are discussed.
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Excision of Skein Categories and Factorisation Homology
TL;DR: In this article, it was shown that the skein categories of Walker-Johnson-Freyd satisfy excision and thus are $k$-linear factorisation homology.
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).