antiselfdual

Antiselfdual, in mathematics, refers to a condition on two-forms or connections in four-dimensional geometry. On an oriented Riemannian 4-manifold, the Hodge star operator maps 2-forms to 2-forms with *^2 = Identity, allowing a decomposition of any 2-form F into self-dual and anti-self-dual parts: F = F^+ + F^-, where F^± = (1/2)(F ± *F). An antiselfdual form satisfies *F = -F, equivalently F^+ = 0.

In gauge theory, the curvature F_A of a connection A on a principal G-bundle is a Lie-algebra–valued 2-form. The antiselfdual condition is *F_A = -F_A, or F_A = F_A^- with F_A^+ = 0. This yields the antiselfdual Yang–Mills equations, a first-order system that implies the Yang–Mills equations and often simplifies the search for solutions.

Antiselfdual connections, also called instantons, are critical points of the Yang–Mills functional and typically minimize its energy within a fixed topological class. On compact 4-manifolds, the moduli space of antiselfdual connections is finite-dimensional under suitable conditions and carries rich geometric and topological information. These moduli spaces underpin Donaldson theory, which uses antiselfdual Yang–Mills invariants to distinguish differentiable structures on 4-manifolds.

Examples include flat connections with zero curvature, and nontrivial instantons on the four-sphere (notably the BPST instanton). The antiselfdual equations are scale-invariant and tightly linked to the broader study of gauge theories in mathematics and theoretical physics.