The Yang-Mills equations over Riemann surfaces
01 Jan 1982-Philosophical Transactions of the Royal Society A (The Royal Society)-Vol. 308, Iss: 1505, pp 523-615
TL;DR: 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.
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TL;DR: In this paper, it was shown that 2+1 dimensional quantum Yang-Mills theory with an action consisting purely of the Chern-Simons term is exactly soluble and gave a natural framework for understanding the Jones polynomial of knot theory in three dimensional terms.
Abstract: It is shown that 2+1 dimensional quantum Yang-Mills theory, with an action consisting purely of the Chern-Simons term, is exactly soluble and gives a natural framework for understanding the Jones polynomial of knot theory in three dimensional terms. In this version, the Jones polynomial can be generalized fromS
3 to arbitrary three manifolds, giving invariants of three manifolds that are computable from a surgery presentation. These results shed a surprising new light on conformal field theory in 1+1 dimensions.
5,093 citations
Cites background from "The Yang-Mills equations over Riema..."
...The topology of Jt is rather intricate (and this was in fact the main subject of interest in [33])....
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...the moduli space of holomorphic vector bundles [33]....
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TL;DR: In this article, a two-parameter generalization of the Seiberg-Witten prepotential is presented, which is rather natural from the M-theory/five dimensional perspective, and conjecture its relation to the tau-functions of KP/Toda hierarchy.
Abstract: Direct evaluation of the Seiberg-Witten prepotential is accomplished following the localization programme suggested in [1]. Our results agree with all low-instanton calculations available in the literature. We present a two-parameter generalization of the Seiberg-Witten prepotential, which is rather natural from the M-theory/five dimensional perspective, and conjecture its relation to the tau-functions of KP/Toda hierarchy.
2,159 citations
Cites background from "The Yang-Mills equations over Riema..."
...6) as a certain topological quantity and apply the powerful equivariant localization techniques [21] to calculate it....
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TL;DR: In this paper, the authors studied a special class of solutions of the self-dual Yang-Mills equations on Riemann surfaces and showed that the moduli space of all solutions turns out to be a manifold with an extremely rich geometric structure.
Abstract: In this paper we shall study a special class of solutions of the self-dual Yang-Mills equations. The original self-duality equations which arose in mathematical physics were defined on Euclidean 4-space. The physically relevant solutions were the ones with finite action-the so-called 'instantons'. The same equations may be dimensionally reduced to Euclidean 3-space by imposing invariance under translation in one direction. These equations also have physical relevance-the solutions which have finite action in three dimensions are the 'magnetic monopoles'. If we take the reduction process one step further and consider solutions which are invariant under two translations, we obtain a set of equations in the plane. Here, however, there is no clear physical meaning and, indeed, attempts to find finite action solutions have failed. Nevertheless, these are the equations we shall consider. Despite the lack of interesting solutions in R2, the equations have the important property-conformal invariance-which allows them to be defined on manifolds modelled on R2 by conformal maps, namely Riemann surfaces. We shall consider here solutions of the self-duality equations defined on a compact Riemann surface. There are in fact solutions, as we shall show, and the moduli space of all solutions turns out to be a manifold with an extremely rich geometric structure which will be the focus of our study. It brings together in a harmonious way the subjects of Riemannian geometry, topology, algebraic geometry, and symplectic geometry. Illuminating all these facets of the same object accounts for the length of this paper. The self-duality equations are equations from gauge theory; geometrically they are defined in terms of connections on principal bundles. While the group of the principal bundle may be chosen arbitrarily for the equations to make sense, we restrict attention here to the simplest case of SU(2) or SO(3). There are two reasons for this. The first, and most obvious, is that it simplifies calculations and avoids the use of inductive processes which are inherent in the consideration of a general Lie group of higher rank. The second reason is that solutions for SU(2) have an intimate relationship with the internal structure of the Riemann surface. As a consequence of results we shall prove about solutions to the self-duality equations, we learn something about the moduli space of complex structures on the surface itself, namely Teichmuller space.
2,047 citations
TL;DR: In this paper, a correspondance entre the geometries algebrique and the geometry differentielle des fibres vectoriels is presented, and a connexion irreductible d'Hermite-Einstein par rapport a metrique ω.
Abstract: On presente une correspondance entre la geometrie algebrique et la geometrie differentielle des fibres vectoriels. Soit une surface algebrique projective X qui a un plongement donne X≤CP N et soit ω une metrique de Kahler sur X dont la classe de cohomologie associee est duale a la classe de section d'hyperplan [H]. Un fibre sur X est stable, par rapport au plongement projectif, si et seulement si il admet une connexion irreductible d'Hermite-Einstein par rapport a la metrique ω. Cette connexion est alors unique
1,295 citations
TL;DR: In this article, the authors propose a solution to solve the problem of spamming, which is called spamming-based spamming.$$$/$/$/$/$$
1,294 citations
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Book•
01 Jan 1978
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.
Abstract: A comprehensive, self-contained treatment presenting general results of the theory. Establishes a geometric intuition and a working facility with specific geometric practices. Emphasizes applications through the study of interesting examples and the development of computational tools. Coverage ranges from analytic to geometric. Treats basic techniques and results of complex manifold theory, focusing on results applicable to projective varieties, and includes discussion of the theory of Riemann surfaces and algebraic curves, algebraic surfaces and the quadric line complex as well as special topics in complex manifolds.
8,196 citations
Book•
06 Apr 2011
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.
Abstract: Introduction.- Doubly Stochastic Matrices.- Schur-Convex Functions.- Equivalent Conditions for Majorization.- Preservation and Generation of Majorization.- Rearrangements and Majorization.- Combinatorial Analysis.- Geometric Inequalities.- Matrix Theory.- Numerical Analysis.- Stochastic Majorizations.- Probabilistic, Statistical, and Other Applications.- Additional Statistical Applications.- Orderings Extending Majorization.- Multivariate Majorization.- Convex Functions and Some Classical Inequalities.- Stochastic Ordering.- Total Positivity.- Matrix Factorizations, Compounds, Direct Products, and M-Matrices.- Extremal Representations of Matrix Functions.
6,641 citations
Book•
01 Jan 1965
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.
Abstract: “Geometric Invariant Theory” by Mumford/Fogarty (the first edition was published in 1965, a second, enlarged edition appeared in 1982) is the standard reference on applications of invariant theory to the construction of moduli spaces. This third, revised edition has been long awaited for by the mathematical community. It is now appearing in a completely updated and enlarged version with an additional chapter on the moment map by Prof. Frances Kirwan (Oxford) and a fully updated bibliography of work in this area. The book deals firstly with actions of algebraic groups on algebraic varieties, separating orbits by invariants and construction quotient spaces; and secondly with applications of this theory to the construction of moduli spaces. It is a systematic exposition of the geometric aspects of the classical theory of polynomial invariants.
2,695 citations
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.
Abstract: We present a self-contained account of the ideas of R. Penrose connecting four-dimensional Riemannian geometry with three-dimensional complex analysis. In particular we apply this to the self-dual Yang-Mills equations in Euclidean 4-space and compute the number of moduli for any compact gauge group. Results previously announced are treated with full detail and extended in a number of directions.
1,574 citations
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).
Abstract: Let X be a compact Riemann surface of genus g _ 2. A holomorphic vector bundle on X is said to be unitary if it arises from a unitary representation of the fundamental group of X. We prove in this paper 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). We recall that Mumford has proved that, on the space of (isomorphic classes) stable vector bundles on X of a given rank and degree, there is a natural structure of a non-singular quasi-projective variety (cf. [7]); further, it was proved in [9] that a vector bundle on X of degree zero is stable if and only if it is associated to an irreducible unitary representation of the fundamental group of X. Thus our result shows the existence of a canonical compactification (as an algebraic variety) of the space of stable bundles on X of a given rank and degree zero. We shall now give a brief outline of the proof. It consists in a refinement of the proof of Mumford for the existence of a natural structure of a quasiprojective variety on the space of stable bundles of a given rank and degree (loc. cit.). Let us fix a very ample invertible sheaf OX(1) on X; then if m is a positive integer which is sufficiently large, we have H'(V(m)) 0 0 and H0( V(m)) generates V(m) for any Ve Or,, where Or, stands for the category of unitary vector bundles on X of rank r. Then the rank of H0(V(m)) is the same whatever be V e OR9; let this be p. The Hilbert polynomial of V(m), is also the same whatever be V e OR,; let this be P. Let Q = Quot(E/P) be the scheme in the sense of Grothendieck; E being the free coherent sheaf of rank p on X (cf. [4]). Let R be the open subscheme of Q consisting of the points which represent quotients of E which are locally free, and whose sections can be canonically identified with H0(E). Thus one has a family of vector bundles {Fq}qeR on X such that every Fq can be canonically considered as a quotient vector bundle of the trivial bundle E on X of rank p. The linear group G = Aut E acts on Q, and R is a G-invariant subscheme; further given V e Or there is a q e R such that Fq V, and the set of such points q con-
1,161 citations