  Lanthanum family of high-temperature cuprate superconductors is known to
exhibit both spin and charge electronic modulations around doping level 1/8. We
assume that these modulations have the character of two-dimensional spin-vortex
checkerboard and investigate whether this assumption is consistent with the
Fermi surface and the pseudogap measured by angle-resolved photo-emission
spectroscopy. We also explore the possibility of observing quantum oscillations
of transport coefficients in such a background. These investigations are based
on a model of non-interacting spin-1/2 fermions hopping on a square lattice and
coupled through spins to a magnetic field imitating spin-vortex checkerboard.
The main results of this article include (i) calculation of Fermi surface
containing Fermi arcs at the positions in the Brillouin zone largely consistent
with experiments; (ii) identification of factors complicating the observations
of quantum oscillations in the presence of spin modulations; and (iii)
investigation of the symmetries of the resulting electronic energy bands,
which, in particular, indicates that each band is double-degenerate and, in
addition, has at least one conical point, where it touches another
double-degenerate band. We discuss possible implications these cones may have
for the transport properties and the pseudogap.
