A topological invariant in the context of the loop representation of the massive Kalb-Ramond-Klein-Gordon model

We employ the Dirac procedure to quantize the self-dual massive Kalb-Ramond-Klein-Gordon model in $2+1$ dimensional spacetimes. The canonical fields are expressed in terms of $2$-surfaces and signed points, ensuring the automatic realization of the quantum algebra. As the duality rotation preserving the action can be implemented infinitesimally, we derive the conserved quantity that generates the transformation. Given that such a generator is a two dimensional topological quantity, its representation in terms of geometrical operators yields a two dimensional invariant (reminiscent of a projection of Gauss’s law in electrodynamics), which encodes the same information of the well-known winding number.