  We introduce a microscopic model aimed at describing the behavior of
fermionic excitations in the background of a magnetic texture called
"spin-vortex checkerboard". This texture was proposed previously as a possible
alternative to stripes to interpret the experimental phenomenology of spin and
charge modulations in 1/8-doped lanthanum cuprates. The model involves two
kinds of interacting fermionic excitations residing in spin-rich and spin-poor
regions of the modulated structure. It is a generalization of another model
developed earlier for the so-called "grid checkerboard". The principal terms of
our model describe the decay of fermionic pairs belonging to spin-poor regions
into single fermions occupying spin-rich regions and vice versa. These terms
induce intricate fermionic correlations throughout the system but fall short of
inducing superconductivity unless arbitrarily small hopping terms are added to
the model Hamiltonian. We present the mean-field solution of the model,
including, in particular, the temperature dependence of the energy gap. The
latter is found to be in a good overall agreement with available experimental
data for high-$T_c$ cuprate superconductors.
