Math 2.0

Global analysis, analysis on manifolds Manifolds of mappings and moduli spaces Object

Moduli space of instantons

Builds on InstantonSpace of connections modulo gaugeGauge group4-manifoldPrincipal bundleChern classModuli space

Description

Let XX be a compact oriented Riemannian 4-manifold and P→XP \to X a principal SU(2)SU(2)-bundle with second Chern class c2(P)=kc_2(P) = k. The moduli space of instantons on PP is ℳk={A:A a connection on P, FA+=0}/𝒢(P),\mathcal{M}_k = \set{A : A \text{ a connection on } P, \ F_A^+ = 0} / \mathcal{G}(P), the set of anti-self-dual connections, the instantons, modulo the gauge group. It is a subset of the space of connections modulo gauge.

Remarks

The linearisation of the equation FA+=0F_A^+ = 0, together with the Coulomb gauge condition, is an elliptic operator. Its index, computed by the Atiyah–Singer index theorem, gives the expected dimension dim⁡ℳk=8k−3(1−b1(X)+b+(X)).\dim \mathcal{M}_k = 8k - 3(1 - b_1(X) + b^+(X)). Here b1b_1 is the first Betti number and b+b^+ the dimension of a maximal positive subspace for the intersection form. Near an irreducible instanton AA at which this linearised operator is surjective, ℳk\mathcal{M}_k is a smooth manifold of this dimension. For X=S4X = S^4 with its round metric and k=1k = 1, the formula gives 8−3=58 - 3 = 5. Indeed ℳ1\mathcal{M}_1 is a 55-dimensional manifold, diffeomorphic to an open 55-ball, and each of its instantons is determined by a centre in S4S^4 and a scale.

The moduli space is not compact in general: instantons can concentrate at points, and Uhlenbeck's compactness theorem describes the limits. Donaldson used these moduli spaces to prove his theorem on definite intersection forms of smooth 44-manifolds and to define new invariants.

Reading path

To understand this page from the ground up, read these 19 pages in order. The foundations come first.

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4-manifold, concept, no evidence level4-manifoldconceptChern class, concept, no evidence levelChern classconceptConnection on a principal bundle, concept, no evidence levelConnection on a princ…conceptCurvature form, concept, no evidence levelCurvature formconceptGroup, concept, no evidence levelGroupconceptInstanton, concept, no evidence levelInstantonconceptLie group, concept, no evidence levelLie groupconceptModuli space, concept, no evidence levelModuli spaceconceptPrincipal bundle, concept, no evidence levelPrincipal bundleconceptQuotient space, concept, no evidence levelQuotient spaceconceptSmooth manifold, concept, no evidence levelSmooth manifoldconceptGroup homomorphism, definition, no evidence levelGroup homomorphismdefinitionSubgroup, definition, no evidence levelSubgroupdefinitionSymmetric group, definition, no evidence levelSymmetric groupdefinitionGauge group, object, no evidence levelGauge groupobjectGroup action, definition, no evidence levelGroup actiondefinitionStabiliser, definition, no evidence levelStabiliserdefinitionSpace of connections modulo gauge, object, no evidence levelSpace of connections …objectModuli space of instantons, object, no evidence levelModuli space of insta…object
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. 4-manifold concept
  2. Chern class concept
  3. Connection on a principal bundle concept
  4. Curvature form concept
  5. Group concept
  6. Instanton concept
  7. Lie group concept
  8. Moduli space concept
  9. Principal bundle concept
  10. Quotient space concept
  11. Smooth manifold concept
  12. Group homomorphism definition
  13. Subgroup definition
  14. Symmetric group definition
  15. Gauge group object
  16. Group action definition
  17. Stabiliser definition
  18. Space of connections modulo gauge object
  19. Moduli space of instantons object

Further reading

  1. Web https://en.wikipedia.org/wiki/Instanton
About this page
TagVP5PW8VSHZ
URIm2://VP5PW8VSHZ
Kindobject
WritingW0 stub
MSC58D27, 53C07, 57R57
Instance ofModuli space
UsesInstanton, Space of connections modulo gauge, Gauge group, 4-manifold, Principal bundle, Chern class, Moduli space

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