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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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On dualizability of braided tensor categories
TL;DR: In this paper, the question of dualizability in higher Morita categories of locally presentable tensor categories and braided tensor classes was studied, and the main result was that the 3-category of rigid tensor category with enough compact projectives is 2-dualizable.
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Homotopical analysis of 4d Chern-Simons theory and integrable field theories
TL;DR: In this paper, a detailed study of $4$-dimensional Chern-Simons theory on arbitrary meromorphic $1$-form $\omega$ on $\mathbb{C}P^1$ using techniques from homotopy theory is investigated.
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Cohomologie des variétés de modules de hauteur nulle
TL;DR: In this article, Atiyah-Bott et al. proposed a cohomologie entiere de M(r,c_1,c-2) for faisceaux stables.
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Unitary representations of Brieskorn spheres
TL;DR: Theorem 3.1 as discussed by the authors implies that any closed connected component of irreducible SU(N) representations of Seifert fibered homology spheres is homeomorphic to a component of an associated genus zero Fuchsian group.
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Cohomology pairings on singular quotients in geometric invariant theory
TL;DR: In this article, the authors give formulas for the pairings of intersection cohomology classes of complementary dimensions of geometric invariant theoretic quotients for which semistability is not necessarily the same as stability.
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).