Journal ArticleDOI
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
Explicit quantization of the Chern-Simons action
TL;DR: In this article, the authors quantized the Chern-Simons action explicitly and found that the geometric quantization of the action strongly depends on the topology of the Riemann surface.
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Convergence properties of the Yang-Mills flow on Kaehler surfaces
TL;DR: In this paper, the Yang-Mills flow on a compact Kahler surface with Kahler form was shown to converge to the double dual of the graded sheaf associated to the Harder-Narasimhan-Seshadri filtration of the holomorphic bundle.
Book ChapterDOI
Wess-Zumino-Witten Conformal Field Theory
TL;DR: In this paper, the Wess-Zumino-Witten (WZW) models of two-dimensional quantum field theory are discussed. But the focus of the course is on the conformal invariant versions of the sigma models with fields taking values in a compact Lie group G.
Journal ArticleDOI
Координаты Дарбу, функционал Янга - Янга и калибровочная теория@@@Darboux coordinates, Yang - Yang functional, and gauge theory
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Langevin dynamic for the 2D Yang-Mills measure
TL;DR: In this article, the authors define a natural state space and Markov process associated to the stochastic Yang-Mills heat flow in two dimensions and construct a framework for applying the theory of regularity structures in the context of vector-valued noise.
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).