Flux tube formation and the Weingarten representation of center vortices and chains

We review some recent results regarding the formulation of mixed ensembles of oriented and nonoriented center vortices based on the Weingarten representation for the sum over surfaces. This framework enabled a partition function for the Abelian-projected ensemble of vortex worldsurfaces, previously formulated as a wavefunctional peaked at center-vortex loops. In particular, we showed how an Abelian ensemble can account for $N$-ality while supporting a ``dual superconductor’’ model for confinement. This formulation also clarified the description of the Goldstone modes for percolating surfaces with non-Abelian degrees of freedom, used in the original mechanism for the formation of a confining flux tube due to the percolating mixed ensemble.