  The population of gamma-ray pulsars, including Crab observed in the TeV
range, and Vela detected above 50 GeV, challenges existing models of pulsed
high-energy emission. Such models should be universally applicable, yet they
should account for spectral differences among the pulsars.
  We show that the gamma-ray emission of Crab and Vela can be explained by
synchrotron radiation from the current sheet of a striped wind, expanding with
a modest Lorentz factor $\Gamma\lesssim100$ in the Crab case, and
$\Gamma\lesssim50$ in the Vela case. In the Crab spectrum a new synchrotron
self-Compton component is expected to be detected by the upcoming experiment
CTA.
  We suggest that the gamma-ray spectrum directly probes the physics of
relativistic magnetic reconnection in the striped wind. In the most energetic
pulsars, like Crab, with $\dot{E}_{38}^{3/2}/P_{-2}\gtrsim0.002$ (where
$\dot{E}$ is the spin down power, $P$ is the pulsar period, and
$X=X_i\times10^i$ in CGS units), reconnection proceeds in the radiative cooling
regime and results in a soft power-law distribution of cooling particles; in
less powerful pulsars, like Vela, particle energization is limited by the
current sheet size, and a hard particle spectrum reflects the acceleration
mechanism. A strict lower limit on the number density of radiating particles
corresponds to emission close to the light cylinder, and, in units of the GJ
density, it is $\gtrsim0.5$ in the Crab wind, and $\gtrsim0.05$ in the Vela
wind.
