Journal ArticleDOI
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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Quotients of unstable subvarieties and moduli spaces of sheaves of fixed Harder–Narasimhan type
Victoria Hoskins,Frances Kirwan +1 more
TL;DR: In this paper, a moduli space of sheaves of fixed Harder-Narasimhan type with some extra data (an '$n$-rigidification') on a projective base is constructed.
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Groupe de picard des variétés de modules de faisceaux semi-stables sur ℙ2
TL;DR: In this paper, a unique fonction δ → ℚ possedant la propriete suivante is defined: for tous r, c1, c2, on a dim(M,c1,c2) > 0 si and seulement si Δ(r,c 1,c 2) ≥ δ( c 1/r).
Journal ArticleDOI
Multiplicative Dirac structures
TL;DR: In this article, the multiplicative Dirac structures on Lie groupoids were introduced, providing a unified framework to study both multiplicative Poisson bivectors and multiplicative closed 2-forms (e.g., symplectic groupoids).
Journal ArticleDOI
The singularities of Yang-Mills connections for bundles on a surface
TL;DR: In this paper, the corresponding Yang-Mills equations for a compact Lie group with Lie algebra g, endowed with an adjoint action invariant scalar product, and a Riemannian metric and orientation on Σ are chosen.
Journal ArticleDOI
Poisson structures on certain moduli spaces for bundles on a surface
TL;DR: In this article, the authors showed that the moduli space of central Yang- Mills connections is stratified by smooth symplectic manifolds and that the holonomy yields a diffeomorphism from moduli spaces to a certain representation space with reference to suitable smooth structures.
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).