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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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Modified Ricci flow on a principal bundle
TL;DR: In this paper, the Ricci Yang-Mills flow was studied in dimension 2 at the Einstein Yang-mills metrics, and it was shown that solutions exist for a short time and are unique.
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Classical Chern-Simons on manifolds with spin structure
TL;DR: In this paper, a 2+1 dimensional classical gauge theory on manifolds with spin structure whose action is a refinement of the Atiyah-Patodi- Singer eta-invariant for twisted Dirac operators was constructed.
Journal ArticleDOI
Geometric prequantization of the moduli space of the vortex equations on a Riemann surface
TL;DR: The moduli space of solutions to the vortex equations on a Riemann surface is well known to have a symplectic (in fact, Kahler) structure as mentioned in this paper, and a family of prequantum line bundles PΩΨ0 on the modulus space whose curvature is proportional to the symplectic forms ΩΩ 0 is known.
Journal ArticleDOI
On the existence of monodromies for the Rabi model
TL;DR: In this article, the existence of monodromies associated with singular points of the eigenvalue problem for the Rabi model is discussed and the complete control of the full monodromy data requires the taming of the Stokes phenomenon associated with the unique irregular singular point.
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The sign problem and the Lefschetz thimble
TL;DR: In this article, the authors proposed a novel approach to deal with the sign problem that hinders Monte Carlo simulations of many quantum field theories (QFTs), which consists in formulating the QFT on a Lefschetz thimble.
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.
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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).