We consider the Regge limit of the CFT correlation functions J J OO and T T OO , where J is a vector current, T is the stress tensor and O is some scalar operator. These correlation functions are related by a type of Fourier transform to the AdS phase shift of the dual 2-to-2 scattering process. AdS unitarity was conjectured some time ago to be positivity of the imaginary part of this bulk phase shift. This condition was recently proved using purely CFT arguments. For large N CFTs we further expand on these ideas, by considering the phase shift in the Regge limit, which is dominated by the leading Regge pole with spin j(ν), where ν is a spectral parameter. We compute the phase shift as a function of the bulk impact parameter, and then use AdS unitarity to impose bounds on the analytically continued OPE coefficients C J J j(ν) and C T T j(ν) that describe the coupling to the leading Regge trajectory of the current J and stress tensor T . AdS unitarity implies that the OPE coefficients associated to non-minimal couplings of the bulk theory vanish at the intercept value ν = 0, for any CFT. Focusing on the case of large gap theories, this result can be used to show that the physical OPE coefficients C J J T and C T T T , associated to non-minimal bulk couplings, scale with the gap ∆ g as ∆ −2 g or ∆ −4 g . Also, looking directly at the unitarity condition imposed at the OPE coefficients C J J T and C T T T results precisely in the known conformal collider bounds, giving a new CFT derivation of these bounds. We finish with remarks on finite N theories and show directly in the CFT that the spin function j(ν) is convex, extending this property to the continuation to complex spin.

In this paper we derive the projectors to all irreducible SO(d) representations (traceless mixed-symmetry tensors) that appear in the partial wave decomposition of a conformal correlator of four stress-tensors in d dimensions. These projectors are given in a closed form for arbitrary length l 1 of the first row of the Young diagram. The appearance of Gegenbauer polynomials leads directly to recursion relations in l 1 for seed conformal blocks. Further results include a differential operator that generates the projectors to traceless mixed-symmetry tensors and the general normalization constant of the shadow operator.

We generalize the embedding formalism for conformal field theories to the case of general operators with mixed symmetry. The index-free notation encoding symmetric tensors as polynomials in an auxiliary polarization vector is extended to mixed-symmetry tensors by introducing a new commuting or anticommuting polarization vector for each row or column in the Young diagram that describes the index symmetries of the tensor. We determine the tensor structures that are allowed in n-point conformal correlation functions and give an algorithm for counting them in terms of tensor product coefficients. A simple derivation of the unitarity bound for arbitrary mixed-symmetry tensors is obtained by considering the conservation condition in embedding space. We show, with an example, how the new formalism can be used to compute conformal blocks of arbitrary external fields for the exchange of any conformal primary and its descendants. The matching between the number of tensor structures in conformal field theory correlators of operators in d dimensions and massive scattering amplitudes in d + 1 dimensions is also seen to carry over to mixed-symmetry tensors.

We propose a method to analytically solve the bootstrap equation for two point functions in boundary CFT. We consider the analytic structure of the correlator in Lorentzian signature and in particular the discontinuity of bulk and boundary conformal blocks to extract CFT data. As an application, the correlator φφ in φ 4 theory at the Wilson-Fisher fixed point is computed to order 2 in the expansion. arXiv:1808.08155v2 [hep-th]