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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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Pontryagin forms on (4r−2)-manifolds and symplectic structures on the spaces of Riemannian metrics
TL;DR: In this paper, the Pontryagin forms on the 1-jet bundle of Riemannian metrics were shown to provide diffeomorphism-invariant pre-symplectic structures for the dimensions n ≡ 2 (mod 4 ).
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On definite lattices bounded by a homology 3-sphere and Yang-Mills instanton Floer theory
TL;DR: In this article, the definite lattices that arise from smooth 4-manifolds bounded by certain homology 3-spheres were determined using instanton Floer theory, extending methods due to Froyshov, and they showed that for +1 surgery on the (2,5) torus knot, the only non-diagonal lattices can occur are E8 and the indecomposable unimodular definite lattice of rank 12, up to diagonal summands.
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The Yang-Mills stratification for surfaces revisited
TL;DR: In this article, Atiyah and Bott's study of Morse theory for the Yang-Mills functional over a Riemann surface was revisited, and new formulas for the minimum codimension of a (non-semi-stable) stratum were established.
Proceedings ArticleDOI
Vector bundles on $p$-adic curves and parallel transport II
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Discreteness for energies of Yang-Mills connections over four-dimensional manifolds
TL;DR: In this article, it was shown that for any principal bundle with compact Lie structure group over a closed, four-dimensional, Riemannian manifold, the energies of Yang-Mills connections on a principal bundle form a discrete sequence without accumulation points.
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).