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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 Li–Yau–Hamilton estimate and the Yang–Mills heat equation on manifolds with boundary
TL;DR: In this article, the Li-Yau-Hamilton estimate for the heat equation on a manifold M with nonempty boundary was established and bounds for a solution ∇ (t) of the Yang-Mills heat equation in a vector bundle over M were established.
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Instantons and singularities in the Yang–Mills flow
TL;DR: In this paper, the authors studied the existence and convergence of the Yang-Mills flow in dimension four and showed that a singularity modeled on an instanton cannot form within finite time.
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The Homotopy Types of $SU(3)$-gauge Groups over Simply Connected 4-manifolds
TL;DR: In this article, the integral homotopy types of gauge groups of principal SU(3)-bundles over a simply-connected Spin 4-manifold M were classified.
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Hermitian Yang-Mills instantons on Calabi-Yau cones
TL;DR: In this article, a non-abelian hermitian Yang-Mills instanton on Calabi-Yau cones was constructed by means of a particular isometry preserving ansatz.
Posted Content
From classical theta functions to topological quantum field theory
Razvan Gelca,Alejandro Uribe +1 more
TL;DR: In this paper, it was shown that the theory of classical theta functions, from the representation theoretic point of view of A. Weil, is just an instance of Chern-Simons theory, and the composition of discrete Fourier transforms and the non-additivity of the signature of 4-dimensional manifolds under gluings obey the same formula.
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.
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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).