Editing Chiral anomaly (section)

===Geometric form===

In the language of [[differential form]]s, to any self-dual curvature form <math>F_A</math> we may assign the abelian 4-form <math>\langle F_A\wedge F_A\rangle:=\operatorname{tr}\left(F_A\wedge F_A\right)</math>.  [[Chern–Weil theory]] shows that this 4-form is locally ''but not globally'' exact, with potential given by the [[Chern–Simons form|Chern–Simons 3-form]] locally:

:<math>d\mathrm{CS}(A)=\langle F_A\wedge F_A\rangle</math>.

Again, this is true only on a single chart, and is false for the global form <math>\langle F_\nabla\wedge F_\nabla\rangle</math> unless the instanton number vanishes.

To proceed further, we attach a "point at infinity" {{math|''k''}} onto <math>\mathbb{R}^4</math> to yield <math>S^4</math>, and use the [[clutching construction]] to chart principal A-bundles, with one chart on the neighborhood of {{math|''k''}} and a second on <math>S^4-k</math>.  The thickening around {{math|''k''}}, where these charts intersect, is trivial, so their intersection is essentially <math>S^3</math>.  Thus instantons are classified by the third [[homotopy group]] <math>\pi_3(A)</math>, which for <math>A = \mathrm{SU(2)}\cong S^3</math> is simply [[Homotopy groups of spheres|the third 3-sphere group]] <math>\pi_3(S^3)=\mathbb{Z}</math>.

The divergence of the baryon number current is (ignoring numerical constants)

:<math>\mathbf{d}\star j_b = \langle F_\nabla\wedge F_\nabla\rangle</math>,

and the instanton number is

:<math>\int_{S^4} \langle F_\nabla\wedge F_\nabla\rangle\in\mathbb{N}</math>.