The Messy Story of Crossing Dispersion Relations

In this talk I will discuss an attempt of unifying under a simple geometric picture, the plethora of crossing symmetric dispersion relations existing in literature.
Each dispersion is valid on a different finite window in energy, but none of them could be extended to infinity.
I will explain how they can all be understood from a simple choice of a foliation of the kinematics space. 
This geometric viewpoint also suggests the existence of a particular (but not so nice) foliation that naturally covers all energies. Unfortunately, this example turns out to be impractical for phenomenological and bootstrap applications. I will comment why, and what teaches us about the dual bootstrap.
 
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