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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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Morse theory for the space of Higgs bundles
TL;DR: In this paper, it was shown that the failure of hyperkähler Kirwan surjectivity for rank 2 fixed determinant Higgs bundles does not occur because of a failure of the existence of a Morse theory.
Journal ArticleDOI
p-Adic strings, the Weil conjectures and anomalies
TL;DR: In this paper, an analogy between the Veneziano amplitude and the p-adic interpolation of the beta function is suggested as the basis of a new padic quantum geometry, and relationships with the Weil conjectures, Fermat curves and anomalies are discussed.
Posted Content
Heat kernel and moduli spaces II
TL;DR: In this article, the moduli spaces of flat G-bundles over a Riemann surface were studied by using heat kernel and Reidemeister torsion, and some general vanishing theorems about characteristic numbers of the modulus spaces were proved.
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Equivariant bifurcation theory and symmetry breaking
TL;DR: In this article, a general genericity and stability theorem for bifurcation diagrams in equivariant bifurbcation theory is proved for all compact Lie groups and absolutely irreducible G-representations.
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Cross ratios, surface groups, PSL(n,ℝ) and diffeomorphisms of the circle
TL;DR: In this paper, the authors identify a connected component of the space of representations into PSL(n,ℝ) known as the n-Hitchin component to a subset of the set of cross ratios on the boundary at infinity of the group.
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).