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=overline{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 Δ 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 ϕ × ϕ OPE must contain at least one primary in the interval [2Δϕ + 2n, 2Δϕ + 2n + 4] for all sufficiently large integer n. The functionals directly compute the OPE decomposition of crossing-symmetrized Witten exchange diagrams in AdS2. 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 AdS2 up to two loops.
Highlights
The conformal bootstrap constitutes a set of powerful nonperturbative constraints on conformal field theories
They are linear functionals acting on the crossing equation which are directly responsible for the optimal bounds of the numerical bootstrap
The equation takes the form aO F∆O (z) = 0, O∈φ×φ where the sum runs over the SL(2) primaries present in the φ × φ OPE and aO ≡2 is the squared OPE coefficient
Summary
The conformal bootstrap constitutes a set of powerful nonperturbative constraints on conformal field theories. We derive useful sum rules satisfied by the CFT data by integrating the crossing equation against appropriate weight-functions in the space of cross-ratios. ∂F∆n(z) stands for the derivative with respect to ∆ of F∆(z) evaluated at ∆ = ∆n This equation tells us that if we add an operator of dimension ∆ to the generalized free solution with a small OPE coefficient, crossing symmetry can be preserved at this order by modifying the scaling dimensions and OPE coefficients of the double traces. The paper is complemented by an appendix containing some technical details
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