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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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Notes on Seiberg-Witten Theory
TL;DR: The Seiberg-Witten invariants of complex surfaces have been studied in this article, where they are invariant to the Gluing technique and to the seibergwitten equation.
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Gromov invariants for holomorphic maps from Riemann surfaces to Grassmannians
TL;DR: In this article, two compactifications of the space of holomorphic maps of fixed degree from a compact Riemann surface to a Grassmannian are studied, and it is shown that the Uhlenbeck compactification has the structure of a projective variety and is dominated by the algebraic compactification coming from the Grothendieck Quot Scheme.
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Boundary value problems for Yang—Mills fields
TL;DR: In this article, the authors investigated boundary value problems for Hermitian Yang-Mills equations over complex manifolds and showed the unique solubility of the Dirichlet problem for the Hermitiansy-mills equation.
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Branes and Quantization
Sergei Gukov,Edward Witten +1 more
TL;DR: In this article, the problem of quantizing a symplectic manifold (M,ω) can be formulated in terms of the A-model of a complexification of M. This leads to an interesting new perspective on quantization.
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Fibrés stables et fibrés exceptionnels sur $\mathbb {P}_2$
J.-M. Drezet,J. Le Potier +1 more
TL;DR: In this paper, the authors determined quelles condi-tions doivent satisfaire r, c 1 and c 2 pour qu'il existe, sur le plan projectif P 2 des fibres vectoriels algebriques stables de rang r de classes de Chern c 1 et c 2.
References
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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.
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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).