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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
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Moduli of vortices and Grassmann manifolds
Indranil Biswas,Nuno M. Romão +1 more
TL;DR: The moduli spaces of stable n-pairs, also interpreted as gauged vortices on a closed Riemann surface with target Mat(r x n, C), where n >= r, are described in this paper.
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Topology of gauge theories on compact 4-manifolds
T P Killingback,E G Rees +1 more
TL;DR: In this article, the topological structure of the group of gauge transformations G can have important consequences and is used to study the existence of a global gauge fixing condition in four-dimensional gauge theories.
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Quantum Yang-Mills Theory in Two Dimensions: Exact versus Perturbative
TL;DR: In this paper, a mathematically rigorous formulation of the perturbative quantization of 2D Yang-Mills was provided, and the asymptotics of exact lattice Wilson loop expectations on $S^2$ were compared with perturbatively computed expectations in holomorphic gauge for simple closed curves to all orders.
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Grafting and Poisson Structure in (2+1)-Gravity with Vanishing Cosmological Constant
TL;DR: In this paper, the authors show how grafting along simple closed geodesics can be implemented in the Chern-Simons formalism and derive explicit expressions for its action on the holonomies of general closed curves on S petertodd g��, where S g�� is an orientable two-surface of genus g>1.
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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.
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).