The Yang-Mills equations over Riemann surfaces

@article{Atiyah1983TheYE,
  title={The Yang-Mills equations over Riemann surfaces},
  author={Michael Francis Atiyah and Raoul Bott},
  journal={Philosophical Transactions of the Royal Society of London. Series A, Mathematical and Physical Sciences},
  year={1983},
  volume={308},
  pages={523 - 615}
}
  • M. Atiyah, R. Bott
  • Published 1983
  • Mathematics
  • Philosophical Transactions of the Royal Society of London. Series A, Mathematical and Physical Sciences
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 ‘perfect' 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… Expand

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References

SHOWING 1-10 OF 45 REFERENCES
On the bundle of connections and the gauge orbit manifold in Yang-Mills theory
In an appropriate mathematical framework we supply a simple proof that the quotienting of the space of connections by the group of gauge transformations (in Yang-Mills theory) is aC∞ principalExpand
Geometry ofSU(2) gauge fields
We studySU(2) Yang-Mills theory onS3×ℝ from the canonical view-point. We use topological and differential geometric techniques, identifying the “true” configuration space as the base-space of aExpand
Self-duality in four-dimensional Riemannian geometry
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-dualExpand
Some remarks on the Gribov ambiguity
The set of all connections of a principal bundle over the 4-sphere with compact nonabelian Lie group under the action of the group of gauge transformations is studied. It is shown that no continuousExpand
Stable and unitary vector bundles on a compact Riemann surface
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 thisExpand
Connections withLP bounds on curvature
We show by means of the implicit function theorem that Coulomb gauges exist for fields over a ball inRn when the integralLn/2 field norm is sufficiently small. We then are able to prove a weakExpand
Principles of Algebraic Geometry
A comprehensive, self-contained treatment presenting general results of the theory. Establishes a geometric intuition and a working facility with specific geometric practices. Emphasizes applicationsExpand
Fixed Points and Torsion on Kahler Manifolds
When a 1-parameter group acts by isometries on a Riemannian manifold M, the fixed point set F is nicely behaved. It is known that each component Fo0 of F is a totally geodesic submanifold of M whoseExpand
Poincaré polynomials of the variety of stable bundles
§1. In this note, we indicate a few improvements to [3]. Let X be an irreducible, non-singular projective algebraic curve defined over a finite field Fq with q elements, of characteristic p. LetExpand
Characteristic classes of stable bundles of rank 2 over an algebraic curve
Let X be a complete nonsingular algebraic curve over C and L a line bundle of degree 1 over X. It is well known that the isomorphism classes of stable bundles of rank 2 and determinant L over X formExpand
...
1
2
3
4
5
...