The Analytic Functional Bootstrap II: Natural Bases for the Crossing Equation

Abstract: We clarify the relationships between different approaches to the conformal bootstrap. A central role is played by the so-called extremal functionals. They are linear functionals acting on the crossing equation which are directly responsible for the optimal bounds of the numerical bootstrap. We explain in detail that the extremal functionals probe the Regge limit. We construct two complete sets of extremal functionals for the crossing equation specialized to $z=\bar{z}$, associated to the generalized free boson and fermion theories. These functionals lead to non-perturbative sum rules on the CFT data which automatically incorporate Regge boundedness of physical correlators. The sum rules imply universal properties of the OPE at large $\Delta$ in every unitary solution of SL(2) crossing. In particular, we prove an upper and lower bound on a weighted sum of OPE coefficients present between consecutive generalized free field dimensions. The lower bound implies the $\phi\times\phi$ OPE must contain at least one primary in the interval $[2\Delta_{\phi}+2n,2\Delta_{\phi}+2n+4]$ for all sufficiently large integer $n$. The functionals directly compute the OPE decomposition of crossing-symmetrized Witten exchange diagrams in $AdS_2$. Therefore, they provide a derivation of the Polyakov bootstrap for SL(2), in particular fixing the so-called contact-term ambiguity. We also use the resulting sum rules to bootstrap several Witten diagrams in $AdS_2$ up to two loops.