We explain the hour-glass magnetic dispersion in underdoped cuprates based on the Lifshitz-type magnetic criticality. While superconductivity influences some details of the dispersion generally it is irrelevant to the hour-glass.
We consider six major experimental observations related to the hour-glass and explain all of them. The theory provides a unified picture of the evolution of magnetic excitations in various cuprate families, including the hour-glass and an emergent static magnetic incommensurate order. We propose the Lifshitz spin liquid sum rule, and show that the data are consistent with the sum rule.

Keywords: cuprate, hour-glass, spin liquid, magnetic criticality