Instanton number - Imported Knowlpedia Content

Definition

Let $E\to M$ be a rank- $r$ Hermitian vector bundle with structure group $SU(r)$ over a closed oriented four-manifold, and let $A$ be a unitary connection with curvature $F_A$ . Its instanton number is

k(E)= -\frac{1}{8\pi^2}\int_M\operatorname{tr}(F_A\wedge F_A) =\langle c_2(E),[M]\rangle\in\mathbb Z,

using the defining representation and the stated sign convention. It is the second Chern number of $E$ , so it is independent of $A$ . When $A$ is a Yang–Mills instanton, this integer labels its topological sector but is not additional connection data.

Chern–Weil independence

The four-form

-\frac{1}{8\pi^2}\operatorname{tr}(F_A\wedge F_A)

represents $c_2(E)$ because $c_1(E)=0$ for an $SU(r)$ -bundle. Replacing $A$ by another connection changes this form by an exact transgression form, whose integral over closed $M$ vanishes. Thus the number depends on the bundle and orientation, not on the chosen representative connection.

For a general compact structure group, an “instanton number” requires choosing an invariant quadratic form normalized to represent an integral characteristic class. The resulting charge need not use the displayed trace normalization.

Energy and self-duality

In four dimensions, decompose $F_A=F_A^++F_A^-$ using the Hodge star. The Chern–Weil integral is proportional to

\|F_A^-\|_{L^2}^2-\|F_A^+\|_{L^2}^2

with the present orientation and trace conventions. The Yang–Mills energy is the corresponding sum. Hence self-dual or anti-self-dual connections saturate a topological lower bound proportional to $|k(E)|$ Donaldson–Kronheimer, §2.1.

Reversing the orientation interchanges self-duality and anti-self-duality and changes the sign of the integral, while leaving its absolute value unchanged.

References

Simon K. Donaldson and Peter B. Kronheimer, The Geometry of Four-Manifolds, Oxford University Press, 1990. DOI record. Relevant: §2.1, characteristic number, curvature decomposition, and the instanton energy bound.
Daniel S. Freed and Karen K. Uhlenbeck, Instantons and Four-Manifolds, 2nd ed., Springer, 1991. DOI record. Relevant: chapter 2, Chern–Weil normalization and topological charge.