Regular ASD connection - Imported Knowlpedia Content

Definition

Let $A$ be an anti-self-dual connection on a principal bundle over a closed oriented Riemannian four-manifold. In its ASD deformation complex, let

H_A^2=\operatorname{coker}\!\left( d_A^+:\Omega^1(X;\operatorname{ad}P)\to \Omega^{2,+}(X;\operatorname{ad}P)\right).

The connection $A$ is regular if $H_A^2=0$ . Equivalently, after Sobolev completion, the linearization $d_A^+$ of the ASD equation is surjective. Regularity therefore says that the linearized equation has no obstruction space. It does not assert that $A$ is irreducible: regularity concerns the cokernel of the linearized equation, whereas irreducibility concerns the stabilizer of the gauge action.

Local consequence

Put the nearby connections $A+a$ in Coulomb gauge $d_A^*a=0$ . The ASD equation becomes

d_A^+a+(a\wedge a)^+=0.

When $A$ is regular, the derivative of this equation is surjective, so the Banach-space implicit-function theorem makes the gauge-fixed solution set smooth near $A$ . If $A$ is also irreducible, the gauge quotient is locally a smooth manifold near $[A]$ , with tangent space $H_A^1$ and the expected dimension Donaldson–Kronheimer, §§4.2–4.3.

Nonregular connections

When $H_A^2\ne0$ , a finite-dimensional Kuranishi map from a neighborhood of $0\in H_A^1$ to $H_A^2$ models the local ASD moduli space. Its zero set can be singular even if $A$ is irreducible. Thus “nonregular” means that the direct implicit-function-theorem argument fails, not that every nearby moduli point must be singular.

Metric dependence and terminology

Regularity depends on the Riemannian metric because the splitting of two-forms and the operator $d_A^+$ do. For suitable generic metrics, transversality theorems often make relevant irreducible ASD connections regular, subject to hypotheses on the bundle and four-manifold.

Some sources call $A$ unobstructed rather than regular. Others reserve “regular moduli space” for a locus on which every connection is both irreducible and unobstructed; that is stronger than the pointwise definition here.

References

Simon K. Donaldson and Peter B. Kronheimer, The Geometry of Four-Manifolds, Oxford University Press, 1990. DOI record. Relevant: §§4.2–4.3, regularity, Kuranishi models, and local dimensions.
Daniel S. Freed and Karen K. Uhlenbeck, Instantons and Four-Manifolds, 2nd ed., Springer, 1991. DOI record. Relevant: Chapter 3, transversality and regular instanton moduli.