We revise the physics of primary electron acceleration in the "slot gap" (SG) above the pulsar polar caps (PCs), a regime originally proposed by Arons & Scharlemann in their electrodynamic model of pulsar PCs. We employ the standard definition of the SG as a pair-free space between the last open field lines and the boundary of the pair plasma column that is expected to develop above the bulk of the PC. The rationale for our revision is that the proper treatment of primary acceleration within the pulsar SGs should take into account the effect of the narrow geometry of the gap on the electrodynamics within the gap and also include the effect of inertial frame dragging on the particle acceleration. We show that the accelerating electric field within the gap, being significantly boosted by the effect of frame dragging, becomes reduced because of the gap geometry by a factor proportional to the square of the SG width. The combination of the effects of frame dragging and geometrical screening in the gap region naturally gives rise to a regime of extended acceleration, which is not limited to "favorably curved" field lines as in earlier models, and the possibility of multiple-pair production by curvature photons at very high altitudes, up to several stellar radii. We present our estimates of the characteristic SG thickness across the PC, energetics of primaries accelerated within the gap, high-energy bolometric luminosities emitted from the high altitudes in the gaps, and maximum heating luminosities produced by positrons returning from the elevated pair fronts. The estimated theoretical high-energy luminosities are in good agreement with the corresponding empirical relationships for γ-ray pulsars. We illustrate the results of our modeling of the pair cascades and γ-ray emission from the high altitudes in the SG for the Crab pulsar. The combination of the frame-dragging field and high-altitude SG emission enables both acceleration at the smaller inclination angles and a larger emission beam, both necessary to produce widely spaced double-peaked profiles.