Non-Euclidean geometry, nontrivial topology and quantum vacuum effects

Space out of a topological defect of the Abrikosov-Nielsen-Olesen vortex type is locally flat but non-Euclidean. If a spinor field is quantized in such a space, then a variety of quantum effects is induced in the vacuum. Basing on the continuum model for long-wavelength electronic excitations, originating in the tight-binding approximation for the nearest neighbor interaction of atoms in the crystal lattice, we consider quantum ground state effects in monolayer structures warped into nanocones by a disclination; the nonzero size of the disclination is taken into account, and a boundary condition at the edge of the disclination is chosen to ensure self-adjointness of the Dirac-Weyl Hamiltonian operator. In the case of carbon nanocones, we find circumstances when the quantum ground state effects are independent of the boundary parameter and the disclination size.