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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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Continuous trace c ∗ -algebras, gauge groups and rationalization
TL;DR: In this article, the rational homotopy group of a principal matrix bundle over a compact metric space was studied and the rational cohomology of the matrix bundle was determined as an H-space of the group of unitaries of the bundle.
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On the Cohomology of Moduli of Vector Bundles and the Tamagawa Number of SL n
TL;DR: In this article, the moduli space of stable rank, degree vector bundles on a smooth projective curve has been computed using a functorial mixed Hodge structure, and the Betti number has been shown to be one.
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A residue theorem for rational trigonometric sums and Verlinde's formula
TL;DR: In this paper, a compact formula for rational trigonometric sums is presented for the dimension of conformal blocks in WZW theory, which coincides with Verlinde's expression.
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Perspectives on geometric analysis
TL;DR: In this paper, a survey paper on several aspects of differential geometry for the last 30 years, especially in those areas related to non-linear analysis, is presented and dedicated to the memory of my teacher S.S. Chern who had passed away in December 2004.
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The Hamiltonian geometry of the space of unitary connections with symplectic curvature
TL;DR: In this article, a moment map interpretation of the Calabi conjecture is presented for the case of a Hermitian line bundle over a compact manifold, where the volume form of a symplectic structure can be seen as finding a zero of the moment map.
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).