An extended gamma-ray emission has been recently discovered by the
HAWC experiment around few Galactic pulsars.
These gamma-ray halos are interpreted as inverse Compton scattering emission
from electrons and positrons accelerated by the pulsar wind nebulae off
Their characterization plays a crucial role in the understanding of particle
acceleration and propagation in pulsars and their nebulae.
In particular, gamma-ray measurements can be used to estimate the contribution
of pulsars to the cosmic-ray positron flux at Earth.
We will discuss a detailed study of the gamma-ray halos around two nearby
pulsars, Geminga and Monogem, and report the first search for their
counterparts in Fermi-LAT data.
We will demonstrate that using gamma-ray data from the LAT is of central
importance to provide a precise estimate for the contribution of these pulsars
to the local cosmic-ray positron flux.