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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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Cohomology of Moduli Spaces
TL;DR: The cohomology of moduli spaces of curves has been studied in this article, where Madsen and Weiss have proposed a proof of a generalisation of Mumford's conjecture.
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Asymptotic faithfulness of the quantum SU(n) representations of the mapping class groups
TL;DR: In this paper, it was shown that the projective representations of the mapping class group obtained from the SU(n)-Verlinde bundles over Teichmuller space are asymptotically faithful, that is the intersection over all levels of the kernels of these representations is trivial, whenever the genus of the underlying surface is at least 3.
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SO(p,q)-Higgs bundles and higher Teichm\"uller components
Marta Aparicio-Arroyo,Steven B. Bradlow,Brian Collier,Oscar García-Prada,Peter B. Gothen,André Oliveira +5 more
TL;DR: In this article, the authors describe new examples of exotic components in moduli spaces of SO(p,q)-Higgs bundles on closed Riemann surfaces and discuss how these exotic components are related to the notion of positive Anosov representations recently developed by Guichard and Wienhard.
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Symplectic and Poisson Geometry of the Moduli Spaces of Flat Connections Over Quilted Surfaces
David Li-Bland,Pavol Ševera +1 more
TL;DR: In this article, a new form of moment-map reduction in the context of Dirac geometry is proposed, where the structure group varies from region to region in the surface, and where a reduction (or relation) of structure occurs along the boundaries of the regions.
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Banach space valued cauchy–riemann equations with totally real boundary conditions
TL;DR: In this article, the authors give a general regularity result for Cauchy-Riemann equations in complex Banach spaces with totally real boundary conditions, which is the first step in a program by Salamon for the proof of the Atiyah-Floer conjecture.
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).