On a 4 dimensional variety there can exist two distinct odd-dimensional
topological period integrals:
The first is the integral of a closed 1-form integrated over a closed
The other is the integral of a closed 3-form integrated over a closed 3
The first idea is at the heart of what has been called the Bohm-Aharanov effect
(or the Berry phase, or the flux quantum).
The other object is the Hopf Index (or the Helicity invariant, or the Knot
invariant, or the Topological Torsion invariant).
As the Helicity invariant is non-zero only in domains of non-uniqueness, its
existence can be interpreted as a defect separating different topological or
thermodynamic phases. For example, the Spinodal line of a Van der Waals gas is
an edge of regression in the Gibbs surface. An edge of regression is where two
surfaces join in a cuspoidal singularity. The edges of the swallowtail
singularity are an edge of regression. The Binodal line is another line of
non-uniqueness and represents a self-intersection type of singularity.
Non-uniqueness implies that A^dA is non-zero.