Anti-self-dual moduli space - Imported Knowlpedia Content

Definition

Let $P\to X$ be a principal bundle with compact structure group over an oriented Riemannian four-manifold. Write $\mathcal A(P)$ for its connections and $\mathcal G(P)$ for its gauge group. The anti-self-dual moduli space is

\mathcal M_{\mathrm{ASD}}(P) =\{A\in\mathcal A(P):F_A^+=0\}/\mathcal G(P),

where $F_A^+$ is the self-dual component of the curvature. Thus its points are gauge-equivalence classes of anti-self-dual connections, or instantons in the common orientation convention. Analytic constructions use compatible Sobolev completions of $\mathcal A(P)$ and $\mathcal G(P)$ , while retaining the same smooth solutions by elliptic regularity.

Local deformation theory

At an ASD connection $A$ , the ASD deformation complex has cohomology $H_A^0,H_A^1,H_A^2$ . Infinitesimal stabilizers form $H_A^0$ , infinitesimal deformations modulo gauge form $H_A^1$ , and $H_A^2$ contains the obstructions. If $A$ is irreducible and regular, a neighborhood of $[A]$ is a smooth manifold with tangent space $H_A^1$ and dimension equal to the index of the deformation complex Donaldson–Kronheimer, §§4.2–4.3.

Singularities and compactification

Irreducibility and regularity are independent conditions. Reducible connections create stabilizer singularities, while nonregular irreducible connections can have obstruction singularities. Even a smooth ASD moduli space need not be compact: sequences can concentrate curvature at finitely many points. Uhlenbeck compactness enlarges it by ideal instantons with lower instanton number plus bubbling points.

Orientation convention

Reversing the orientation of $X$ exchanges self-dual and anti-self-dual two-forms and hence exchanges the ASD and self-dual moduli spaces. Some authors use “instanton moduli space” for $F_A^+=0$ , others for $F_A^-=0$ . The displayed equation removes this ambiguity.

The ASD moduli space is a distinguished subspace of the full Yang–Mills moduli space, because every ASD connection is Yang–Mills but not every Yang–Mills connection is ASD.

References

Simon K. Donaldson and Peter B. Kronheimer, The Geometry of Four-Manifolds, Oxford University Press, 1990. DOI record. Relevant: Chapter 4, ASD moduli spaces, deformation theory, and compactification.
Daniel S. Freed and Karen K. Uhlenbeck, Instantons and Four-Manifolds, 2nd ed., Springer, 1991. DOI record. Relevant: Chapter 3, gauge quotients and local instanton moduli.