  The magnetosphere of a rotating pulsar naturally develops a current sheet
beyond the light cylinder (LC). Magnetic reconnection in this current sheet
inevitably dissipates a nontrivial fraction of the pulsar spin-down power
within a few LC radii. We develop a basic physical picture of reconnection in
this environment and discuss its implications for the observed pulsed gamma-ray
emission. We argue that reconnection proceeds in the plasmoid-dominated regime,
via an hierarchical chain of multiple secondary islands/flux ropes. The
inter-plasmoid reconnection layers are subject to strong synchrotron cooling,
leading to significant plasma compression. Using the conditions of pressure
balance across these current layers, the balance between the heating by
magnetic energy dissipation and synchrotron cooling, and Ampere's law, we
obtain simple estimates for key parameters of the layers --- temperature,
density, and layer thickness. In the comoving frame of the relativistic pulsar
wind just outside of the equatorial current sheet, these basic parameters are
uniquely determined by the strength of the reconnecting upstream magnetic
field. For the case of the Crab pulsar, we find them to be of order 10 GeV,
$10^{13} cm^{-3}$, and 10 cm, respectively. After accounting for the bulk
Doppler boosting due to the pulsar wind, the synchrotron and inverse-Compton
emission from the reconnecting current sheet can explain the observed pulsed
high-energy (GeV) and VHE (~100 GeV) radiation, respectively. Also, we suggest
that the rapid relative motions of the secondary plasmoids in the hierarchical
chain may contribute to the production of the pulsar radio emission.
