  We study the origin of the positron excess observed in the local cosmic-ray
spectrum at high energies, and relate it to the cosmic rays and gamma-ray
emission across the entire Galaxy. In particular, we explore the hypothesis of
a single, dominant source accountable for primary electron-positron pairs.
Since we are agnostic about the physical nature of the underlying source
population, we consider four simple models that are representative of young
pulsars, old stars (as a tracer of millisecond pulsars), and annihilating dark
matter particles. In the dark matter hypothesis, we consider both a cored and a
cuspy model for the halo in the Milky Way. Then, we compare the associated
gamma-ray sky maps with Fermi-LAT data. The aim of this work is not to derive
constraints or upper limits for the different models considered, but rather to
explore the possibility, as a proof of concept, of building a self-consistent
model able to explain simultaneously the origin of all cosmic-ray species,
including positrons, as well as the Galactic center GeV gamma-ray emission. We
find that the emission arising from pulsar wind nebulae is fairly concentrated
near the mid plane, and therefore additional cosmic-ray sources must be invoked
to explain the emission at the center of the Galaxy. If the local positron
excess were mainly due to millisecond pulsars, inverse Compton scattering by
the particles injected in the Milky Way bulge would naturally account for a
non-negligible fraction of the central gamma-ray emission. The case of
annihilating dark matter is very sensitive to the precise shape of the dark
matter profile. The results for a standard NFW cuspy profile are above the
gamma-ray measurements by as much as a factor of 2 in some regions of the
Galaxy, while the results for an isothermal, cored profile are still compatible
with the data. However, the cross-sections exceed the current constraints.
