Math 2.0

Global analysis, analysis on manifolds Theorem

Atiyah–Bott theorem on Yang–Mills over Riemann surfaces

Builds on Yang–Mills functionalYang–Mills equationsRiemann surfacePrincipal bundleConnection on a principal bundleCurvature formHolomorphic vector bundleStable vector bundleEquivariant cohomologyGauge groupModuli space

Statement

Let MM be a closed Riemann surface of genus gg with a compatible Riemannian metric and area form ω\omega. Let P→MP \to M be a principal U(n)U(n)-bundle of degree dd, let 𝒜\mathcal{A} be the space of connections on PP, and let 𝒢\mathcal{G} be its gauge group. Each A∈𝒜A \in \mathcal{A} defines a holomorphic structure EAE_A on the associated vector bundle EE of rank nn.

  1. A connection AA satisfies the Yang–Mills equations if and only if EE splits in the following way. There is an orthogonal direct sum E=E1⊕⋯⊕ErE = E_1 \oplus \cdots \oplus E_r of subbundles preserved by AA. On each EjE_j the curvature of AA equals cj idEj⊗ωc_j \, \mathrm{id}_{E_j} \otimes \omega for a constant cjc_j.
  2. The stratification of 𝒜\mathcal{A} by the Harder–Narasimhan type of EAE_A is 𝒢\mathcal{G}-equivariantly perfect for rational coefficients. So the Yang–Mills functional behaves on 𝒜\mathcal{A} like an equivariantly perfect Morse function.
  3. Let nn and dd be coprime. The moduli space of stable bundles of rank nn and degree dd on MM has integral cohomology without torsion. Its Poincaré polynomial is determined by an explicit recursion over Harder–Narasimhan types.

Remarks

On a surface, the Yang–Mills equation dA∗FA=0d_A^* F_A = 0 says that the function ⋆FA\star F_A, a section of the adjoint bundle, is covariantly constant. Its eigenvalues are therefore constant and its eigenspaces give the parallel splitting of part 1. By the Narasimhan–Seshadri theorem, the connections of constant central curvature correspond to polystable holomorphic bundles, so the critical sets are described by stable bundles of lower rank.

For part 2 one compares the equivariant cohomology of 𝒜\mathcal{A}, which is the cohomology of the classifying space of 𝒢\mathcal{G} because 𝒜\mathcal{A} is contractible, with that of the strata. Perfection means that the Thom–Gysin sequences of the strata split into short exact sequences. The open stratum consists of the semistable bundles, and in the coprime case the gauge group acts on it with constant stabiliser U(1)U(1), which gives part 3.

Going deeper

For a student

The work links gauge theory on surfaces with algebraic geometry of vector bundles on curves through the Narasimhan–Seshadri theorem. It is a model of infinite-dimensional Morse theory with a symmetry group, carried out through equivariant cohomology. The Harder–Narasimhan stratification it uses also appears in moment map geometry, where the curvature plays the role of a moment map for the gauge group.

Reading path

To understand this page from the ground up, read these 14 pages in order. The foundations come first.

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Connection on a principal bundle, concept, no evidence levelConnection on a princ…conceptCurvature form, concept, no evidence levelCurvature formconceptEquivariant cohomology, concept, no evidence levelEquivariant cohomologyconceptHolomorphic vector bundle, concept, no evidence levelHolomorphic vector bu…conceptLie group, concept, no evidence levelLie groupconceptModuli space, concept, no evidence levelModuli spaceconceptPrincipal bundle, concept, no evidence levelPrincipal bundleconceptRiemann surface, concept, no evidence levelRiemann surfaceconceptSmooth manifold, concept, no evidence levelSmooth manifoldconceptStable vector bundle, concept, no evidence levelStable vector bundleconceptYang–Mills equations, concept, no evidence levelYang–Mills equationsconceptYang–Mills functional, concept, no evidence levelYang–Mills functionalconceptGauge group, object, no evidence levelGauge groupobjectAtiyah–Bott theorem on Yang–Mills over Riemann surfaces, statement, no evidence levelAtiyah–Bott theorem o…statement
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
  • no evidence level
  1. Connection on a principal bundle concept
  2. Curvature form concept
  3. Equivariant cohomology concept
  4. Holomorphic vector bundle concept
  5. Lie group concept
  6. Moduli space concept
  7. Principal bundle concept
  8. Riemann surface concept
  9. Smooth manifold concept
  10. Stable vector bundle concept
  11. Yang–Mills equations concept
  12. Yang–Mills functional concept
  13. Gauge group object
  14. Atiyah–Bott theorem on Yang–Mills over Riemann surfaces statement

Further reading

  1. Web https://en.wikipedia.org/wiki/Yang–Mills_equations
About this page
TagAV0Q1VSJ88
URIm2://AV0Q1VSJ88
Kindstatement, theorem
WritingW0 stub
MSC58E15, 53C07, 14H60
StatusThis copy of the site has no build data. Run m2 status AV0Q1VSJ88.
AboutYang–Mills functional, Yang–Mills equations
UsesYang–Mills functional, Yang–Mills equations, Riemann surface, Principal bundle, Connection on a principal bundle, Curvature form, Holomorphic vector bundle, Stable vector bundle, Equivariant cohomology, Gauge group, Moduli space

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