Spin current in BF theory

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.