  We derive new crossing-symmetric dispersion formulae for CFT correlators
restricted to the line. The formulae are equivalent to the sum rules implied by
what we call master functionals, which are analytic extremal functionals which
act on the crossing equation. The dispersion relations provide an equivalent
formulation of the constraints of the Polyakov bootstrap and hence of crossing
symmetry on the line. The built in positivity properties imply simple and exact
lower and upper bounds on the values of general CFT correlators on the
Euclidean section, which are saturated by generalized free fields. Besides
bounds on correlators, we apply this technology to determine new universal
constraints on the Regge limit of arbitrary CFTs and obtain very simple and
accurate representations of the 3d Ising spin correlator.
