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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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The complex symplectic geometry of the deformation space of complex projective structures
TL;DR: In this article, the complex symplectic geometry of the deformation space of complex projective structures on a closed oriented surface of genus at least 2 is investigated and compared to the Goldman symplectic structure on the character variety, clarifying and generalizing a theorem of S Kawai.
Journal ArticleDOI
Equivariant classes of matrix matroid varieties
TL;DR: In this paper, the equivariant cohomology class represented by the Zariski closure Y = X is studied and the coefficients of this class are solutions to problems in enumerative geometry, which are natural generalization of the linear Gromov-Witten invariants of projective spaces.
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Residue theorem for rational trigonometric sums and Verlinde's formula
TL;DR: In this paper, a compact formula for rational trigonometric sums appeared in the work of E. Verlinde on the dimension of conformal blocks in Wess-Zumino-Witten (WZW) theory.
Journal ArticleDOI
Абелева лагранжева алгебраическая геометрия@@@Abelian Lagrangian algebraic geometry
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A Quantum group structure in integrable conformal field theories
TL;DR: In this paper, a quantization prescription of conformal algebras of a class of d = 2 conformal field theories which are integrable is discussed, and the condition under which the representations of these quantum groups admit a Hilbert space completion by exploring the relation with the braid group.
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).