Vacuum structure of toroidal carbon nanotubes

Low-energy excitations in carbon nanotubes can be described by an effective field theory of two-component spinors. It is pointed out that the chiral anomaly in $1+1$ dimensions should be observed in a metallic toroidal carbon nanotube on a planar geometry with varying magnetic field. We also analyze the vacuum structure of the metallic toroidal carbon nanotube including the one-dimensional long-range Coulomb interactions and discuss some effects of external charges on the vacuum.