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Manifolds and cell complexes Definition

Donaldson theory

Builds on 4-manifoldSmooth manifoldRiemannian metricPrincipal bundleConnection on a principal bundle

Definition

Let MM be a closed, oriented, smooth 4-manifold with a Riemannian metric gg, and let ∗\ast be the Hodge star of gg on 22-forms. Let P→MP \to M be a principal bundle with structure group SU(2)SU(2) and second Chern class c2(P)=kc_2(P) = k. For a connection AA on PP with curvature FAF_A, write FA=FA++FA−F_A = F_A^+ + F_A^- with ∗FA±=±FA±\ast F_A^{\pm} = \pm F_A^{\pm}.

  1. The connection AA is anti-self-dual if FA+=0F_A^+ = 0.
  2. The gauge group 𝒢\mathcal{G} of bundle automorphisms of PP acts on anti-self-dual connections.
  3. The moduli space ℳk(g)\mathcal{M}_k(g) is the set of anti-self-dual connections on PP modulo 𝒢\mathcal{G}.

Donaldson theory is the study of the spaces ℳk(g)\mathcal{M}_k(g) and of the invariants of MM obtained from them. The main such invariants are the Donaldson polynomial invariants, defined by integrating cohomology classes over ℳk(g)\mathcal{M}_k(g).

Remarks

Let b1(M)b_1(M) be the first Betti number and b2+(M)b_2^+(M) the number of positive eigenvalues of the intersection form. For a generic metric, the irreducible part of ℳk(g)\mathcal{M}_k(g) is a smooth manifold of dimension 8k−3(1−b1(M)+b2+(M))8k - 3(1 - b_1(M) + b_2^+(M)). For example, for k=1k = 1 and MM simply connected with negative definite form, the dimension is 8−3=58 - 3 = 5; this is the moduli space used in the proof of Donaldson's theorem. A connection with FA−=0F_A^- = 0 but FA+≠0F_A^+ \ne 0 is self-dual, not anti-self-dual, and so does not lie in ℳk(g)\mathcal{M}_k(g); reversing the orientation of MM exchanges the two notions. Anti-self-dual connections are the absolute minima of the Yang–Mills functional in their bundle. The polynomial invariants are defined when b2+(M)≥2b_2^+(M) \ge 2 and do not depend on the metric.

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Further reading

  1. Web https://en.wikipedia.org/wiki/Donaldson_theory
About this page
TagDRNH767W84
URIm2://DRNH767W84
Kinddefinition
WritingW0 stub
MSC57R57, 57K41, 58D27
DefinesDonaldson theory
Uses4-manifold, Smooth manifold, Riemannian metric, Principal bundle, Connection on a principal bundle
Changed2 November 2026

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