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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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The Moduli of Flat PU(p,q) Structures on Riemann Surfaces
Eyal Markman,Eugene Z. Xia +1 more
TL;DR: In this paper, it was shown that the Toledo invariant associated with each element in Hom+(π1(X), U(p, 1))/U(p/p/1) is 2(g−1)+1.
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Continuous smearing of Wilson Loops
TL;DR: Continuum smearing was introduced in section 4.1 of JHEP03, 064 (2006) as a meaningful continuum analogue of the well known set of lattice techniques by the same name as discussed by the authors.
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A perturbation method for spinorial Yamabe type equations on \(S^m\) and its application
TL;DR: In this paper, the existence of non-trivial solutions for Dirac equations with critical Sobolev nonlinearities via a perturbative variational method was proved for the special case (m = 2).
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Connected Components of The Space of Surface Group Representations
Nan-Kuo Ho,Chiu-Chu Melissa Liu +1 more
TL;DR: Alekseev, Malkin and Meinrenken as discussed by the authors showed that a closed compact nonorientable surface is homeomorphic to the connected sum of k copies of the real projective plane.
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Real loci of symplectic reductions
Rebecca Goldin,Tara S. Holm +1 more
TL;DR: In this paper, the Kirwan surjectivity theorem was extended to real loci, and the kernel of the real Kirwan map was computed for real locus reduction with non-natural Hamiltonian torus actions.
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.
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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).