Knowlpedia

Definition

Yang–Mills instanton

A finite-action self-dual or anti-self-dual connection on an oriented Riemannian four-manifold.

Definition

Let P→MP\to M be a with compact structure group over an oriented Riemannian four-manifold, and choose an invariant on its . A Yang–Mills instanton is a AA whose has finite action

∫M∣FA∣2 dvol⁡<∞\int_M |F_A|^2\,d\operatorname{vol}<\infty

and satisfies one of the first-order equations

∗FA=FAor∗FA=−FA.*F_A=F_A \qquad\text{or}\qquad *F_A=-F_A .

Equivalently, AA is a of finite energy. On compact MM, finiteness is automatic. A common orientation convention reserves “instanton” for the anti-self-dual equation.

Relation to the Yang–Mills equation

The gives dAFA=0d_AF_A=0. If ∗FA=±FA*F_A=\pm F_A, then dA(∗FA)=0d_A(*F_A)=0, so every instanton is a . The converse fails: a Yang–Mills connection can have both self-dual and anti-self-dual curvature components.

On a compact four-manifold, the orthogonal decomposition FA=FA++FA−F_A=F_A^++F_A^- rewrites the Yang–Mills action as a topological Chern–Weil term plus a nonnegative multiple of either ∥FA+∥L22\|F_A^+\|_{L^2}^2 or ∥FA−∥L22\|F_A^-\|_{L^2}^2. Hence instantons attain the absolute energy bound in their fixed topological class Donaldson–Kronheimer, §2.1.

Canonical example

The BPST connection on the charge-one SU(2)SU(2)-bundle over S4S^4 is the basic nonflat instanton. Removing one point and using stereographic coordinates gives a finite-action instanton on R4\mathbb R^4; its curvature decay makes the noncompact action finite. Its self-dual versus anti-self-dual label changes when the orientation is reversed.

A general finite-action Yang–Mills connection with neither FA+=0F_A^+=0 nor FA−=0F_A^-=0 is a near miss: finite action and the second-order do not imply the instanton equation.

Conventions and scope

Some authors exclude flat connections or require nonzero , while the stated definition includes the zero-curvature solution. In higher-dimensional , “instanton” can mean a connection solving a different first-order equation determined by special holonomy or a calibration. Such generalized instantons are not covered by this four-dimensional definition.

References
  1. Simon K. Donaldson and Peter B. Kronheimer, The Geometry of Four-Manifolds, Oxford University Press, 1990. DOI record. Relevant: §2.1, self-duality, the energy identity, and instanton number.
  2. Daniel S. Freed and Karen K. Uhlenbeck, Instantons and Four-Manifolds, 2nd ed., Mathematical Sciences Research Institute Publications 1, Springer, 1991. DOI record. Relevant: “The Yang–Mills Equations,” pp. 28–43, finite-action self-dual connections and their moduli.