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

TL;DR

This paper introduces a topological invariant derived from the loop representation of a 2+1 dimensional massive Kalb-Ramond-Klein-Gordon model, revealing a new geometric quantity related to duality symmetry.

Contribution

It presents a novel topological invariant in the loop representation of the model, connecting duality symmetry to geometric operators and extending understanding of topological aspects in field theories.

Findings

Derived a conserved topological quantity generating duality transformations.

Expressed the invariant in terms of geometrical operators, linking it to a winding number.

Revealed a geometric interpretation similar to a projection of Gauss's law.

Abstract

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.