Relativistic reconnection is thought to be an efficient mechanism of the energy dissipation in magnetically dominated outflows. It has been considered in the context of blazar flares, Crab flares, and also as a process underlying pulsar emission.
I will consider the latter scenario, arguing that the physics of particle acceleration at the reconnection sites in pulsar winds is imprinted on the gamma-ray spectra of Fermi pulsars. In the very most energetic ones, like the Crab, the acceleration is limited by radiative cooling, while in less powerful pulsars, like Vela, the particle energization is limited by the size of the acceleration region -- in this case the reconnecting current sheet.
I will show a model constraint on the Lorentz factor of the wind. The upper limit corresponds to the model of emission generated close to the light cylinder, and it is ~100 in the Crab case and ~50 in the Vela case. I will discuss also predictions for a new spectral component due to synchrotron self-compton process, which extends up to tens of TeV in the spectrum of Crab and which should be detected by the upcoming experiment CTA.