Researchers studying three-dimensional BF theory showed how a doubled Chern–Simons theory at increasingly large level approaches a theory with continuous symmetry. They proved that normalized states on a torus keep converging when fixed changes of the torus are applied.
The result gives a controlled bridge between exact calculations at finite levels and a simpler continuous theory. It also clarifies how magnetic flux and electric charge exchange under a change of basis, while pinning down what can and cannot survive at the limit.
✦ At the boundary between finite and continuous symmetry, flux and charge still behave predictably.