Fermion mass generation and induced current in low-dimensional models with nontrivial topology

We study the process of dynamical fermion mass generation in a three-dimensional model with two types of interacting fermions existing in three dimensions and on a two-dimensional brane with the action of the $A_3$ gauge field taken into account. We investigate the model of a nanotube with the dimension $\mathbb R^2\times S^1$. We establish that in the expression for the fermionic gap, along with the dynamical contribution of the condensate $m$, the four-fermion interaction results in another kinematic contribution, an Aharonov–Bohm phase calculated from the condition of a minimum of the effective potential of the model. We demonstrate the possibility of induced current generation due to the effect of vacuum polarization with the role played by the nontrivial topology of the problem taken into account.