Spin current in BF theory

TL;DR

This paper introduces a spin current concept in BF theory, proposing a new term in the Lagrangian to compute it, and derives solutions in symmetric systems without requiring a specific metric.

Contribution

It presents a novel method to define and calculate spin currents in BF theory using an added field, expanding understanding of solutions without metric constraints.

Findings

Derived equations of motion for spin current in BF theory

Proposed a new Lagrangian term involving a field  solutions in symmetric systems

Abstract

In this paper, we introduce a current which we call spin current corresponding to the variation of the matter action in BF theory with respect to the spin connection $A$ which takes values in Lie algebra $\mathfrak{so}(3,\mathbb{C})$ in self-dual formalism. For keeping the constraint $DB^i=0$ satisfied, we suggest adding a new term to the BF Lagrangian using a new field $\psi^i$ which can be used for calculating the spin current. We derive the equations of motion and discuss the solutions. We will see that the solutions of that equations do not require a specific metric on the manifold $M$, we just need to know the symmetry of the system and the information about the spin current. Finally we find the solutions in a spherical and cylindrical symmetric systems.