Knowlpedia

Definition

Self-dual and anti-self-dual connection

A connection on an oriented Riemannian four-manifold whose curvature lies entirely in the self-dual or anti-self-dual summand of two-forms.

Definition

Let P→MP\to M be a over an oriented Riemannian four-manifold, and let AA be a with FA∈Ω2(M;ad⁡P)F_A\in\Omega^2(M;\operatorname{ad}P). The metric and orientation define a with ∗2=1*^2=1 on two-forms. The connection is self-dual if

∗FA=FA,*F_A=F_A,

and anti-self-dual if

∗FA=−FA.*F_A=-F_A.

Equivalently, FAF_A lies in the +1+1 or −1-1 eigenbundle of the Hodge star. Reversing the orientation interchanges the two conditions.

Why these connections are Yang–Mills

The gives dAFA=0d_AF_A=0. If ∗FA=±FA*F_A=\pm F_A, it follows that dA(∗FA)=0d_A(*F_A)=0, which is the in the convention dA(∗FA)=0d_A(*F_A)=0. Thus every self-dual or anti-self-dual connection is a . The converse is false: the Yang–Mills equation is second order in the connection, whereas the self-duality equations impose the stronger first-order curvature condition Donaldson–Kronheimer, §2.1.

Curvature splitting and examples

The Hodge star splits bundle-valued two-forms orthogonally as

Ω2(M;ad⁡P)=Ω+2(M;ad⁡P)⊕Ω−2(M;ad⁡P).\Omega^2(M;\operatorname{ad}P) =\Omega^2_+(M;\operatorname{ad}P)\oplus \Omega^2_-(M;\operatorname{ad}P).

Writing FA=FA++FA−F_A=F_A^++F_A^-, self-duality means FA−=0F_A^-=0, while anti-self-duality means FA+=0F_A^+=0. Every satisfies both equations because FA=0F_A=0; a nonflat connection cannot satisfy both.

The basic SU(2)SU(2) BPST instanton on R4\mathbb R^4 gives a nonflat solution of one of the two equations. Whether it is labeled self-dual or anti-self-dual depends on the orientation and curvature conventions.

Conventions and scope

The four-dimensional hypothesis is essential to this formulation: only in dimension four does the Hodge star carry two-forms to two-forms and square to +1+1 in Riemannian signature. The equations themselves do not require compactness of MM, but finite-action moduli theory normally imposes compactness or decay conditions and uses an Ad⁡\operatorname{Ad}-invariant on the .

References
  1. Simon K. Donaldson and Peter B. Kronheimer, The Geometry of Four-Manifolds, Oxford University Press, 1990. Publisher record. Relevant: §2.1, self-duality, curvature splitting, and the Yang–Mills functional.
  2. Daniel S. Freed and Karen K. Uhlenbeck, Instantons and Four-Manifolds, 2nd ed., Springer, 1991. Publisher record. Relevant: Chapter 2, the self-dual Yang–Mills equations.