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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 homotopy types of su(5)-gauge groups
TL;DR: In this paper, it was shown that there is a rational or $p$-local homotopy equivalence if and only if the second Chern class is a prime or prime.
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Crossing number of alternating knots in S × I
TL;DR: In this paper, a generalization of the fit conjecture for knots in S × I, where S is a surface, was proved using a combination of geometric and polynomial techniques.
Journal ArticleDOI
p-form electromagnetism on discrete spacetimes *
TL;DR: In this paper, a chain complex equipped with an inner product on (p + 1 )-cochains was constructed for the p-form electromagnetism, with either R or U( 1) as a gauge group.
Posted Content
Hitchin's Projectively Flat Connection, Toeplitz Operators and the Asymptotic Expansion of TQFT Curve Operators
TL;DR: In this article, the authors provide a review of the geometric construction, proposed by Witten, of the SU(n) quantum representations of the mapping class groups which are part of the Reshetikhin-Turaev TQFT for the quantum group U_q(sl(n, C)).
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Mirror Symmetry, Hitchin's Equations, And Langlands Duality
TL;DR: In this paper, the essential statements for the ''unramified'' case that a compact oriented two-manifold without boundary allows punctures are sketched, and a few indications of why a more precise description requires a starting point in four-dimensional gauge theory.
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.
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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).