Here, AiA_i encodes the density/strength of circulation and i\theta_i the angular phase of the superfluid order parameter.
Thus, a network of vortex--antivortex pairs populates the informational manifold with a set of discrete nodes, each endowed with a phase label (holonomy charge).
2. Tunneling as holonomy creation
Vacuum tunneling events in the 2D superfluid correspond to instantaneous creation and annihilation of vortex--antivortex pairs. From the InterTop perspective:
Creation of a vortex pair = emergence of two new informational nodes with opposite phase windings.
Annihilation = holonomy cancellation, where opposite informational charges neutralize.
Crucially, tunneling events do not merely change energy levels; they restructure the connectivity of the informational manifold.
Mathematically, a tunneling event at time t0t_0 inserts a pair of nodes {Nv,Nv}\{\mathcal{N}_v, \mathcal{N}_{\bar v}\} into the manifold with amplitudes
v=Ave+i,v=Avei.\Psi_v = A_v e^{+i\theta}, \quad \Psi_{\bar v} = A_{\bar v} e^{-i\theta}.
Their joint contribution to the manifold cancels globally but produces local holonomy loops around each vortex core.
3. Informational holonomy of vortex circulation
The circulation integral of the superfluid order parameter around a closed loop CC enclosing a vortex is
Cdr=2n,\oint_C \nabla \theta \cdot d\mathbf{r} = 2\pi n,
where nZn \in \mathbb{Z} is the winding number.
