We argue that every CFT contains light-ray operators labeled by a continuous spin J. When J is a positive integer, light-ray operators become integrals of local operators over a null line. However for non-integer J, light-ray operators are genuinely nonlocal and give the analytic continuation of CFT data in spin described by Caron-Huot. A key role in our construction is played by a novel set of intrinsically Lorentzian integral transforms that generalize the shadow transform. Matrix elements of light-ray operators can be computed via the integral of a double-commutator against a conformal block. This gives a simple derivation of Caron-Huot's Lorentzian OPE inversion formula and lets us generalize it to arbitrary fourpoint functions. Furthermore, we show that light-ray operators enter the Regge limit of CFT correlators, and generalize conformal Regge theory to arbitrary four-point functions. The average null energy operator is an important example of a light-ray operator. Using our construction, we find a new proof of the average null energy condition (ANEC), and furthermore generalize the ANEC to continuous spin. H.1.3 Rules for weight-shifting operators 104 H.2 A Lorentzian integral for a conformal block 105 H.2.1 Shadow transform in the diamond 106 H.3 Conformal blocks at large J 107 ∼Here, f 12O (J) and f 34O (J) are OPE coefficients that have been analytically continued in the spin J of O. The parameter t measures the boost of O 1 , O 2 relative to O 3 , O 4 . J 0 ∈ R is the Regge/Pomeron intercept, and is determined by the analytic continuation of the dimension ∆ O to non-integer J. 2 The ". . . " in (1.2) represent higher-order corrections in 1/N 2 and also terms that grow slower than e t(J 0 −1) in the Regge limit t → ∞.A missing link in this story was provided recently by Caron-Huot, who proved that OPE coefficients and dimensions have a natural analytic continuation in spin in any CFT [16]. The analytic continuation of OPE data in a scalar four-point function φ 1 φ 2 φ 3 φ 4 can be computed by a "Lorentzian inversion formula," given by the integral of a double-relative to a conventional conformal block. Caron-Huot's Lorentzian inversion formula has many other useful applications, for example to large-spin perturbation theory and the lightcone bootstrap [17][18][19][20][21][22][23][24][25][26], and to the SYK model [27][28][29][30]. 3 However, Caron-Huot's result raises some obvious questions:• Can operators themselves (not just their OPE data) be analytically continued in spin?• What is the space of continuous spin operators in a given CFT?• Do continuous-spin operators have a Hilbert space interpretation (similar to how integerspin operators correspond to CFT states on S d−1 )?• What is the meaning of the funny block in the Lorentzian inversion formula, and how do we generalize it?Answering these questions is important for making sense of the Regge limit, and more generally for understanding how to write a convergent OPE in non-vacuum states.2 In d = 2, the Regge regime is the same as the chaos regime. ...

We introduce a large class of conformally-covariant differential operators and a crossing equation that they obey. Together, these tools dramatically simplify calculations involving operators with spin in conformal field theories. As an application, we derive a formula for a general conformal block (with arbitrary internal and external representations) in terms of derivatives of blocks for external scalars. In particular, our formula gives new expressions for "seed conformal blocks" in 3d and 4d CFTs. We also find simple derivations of identities between external-scalar blocks with different dimensions and internal spins. We comment on additional applications, including deriving recursion relations for general conformal blocks, reducing inversion formulae for spinning operators to inversion formulae for scalars, and deriving identities between general 6j symbols (Racah-Wigner coefficients/"crossing kernels") of the conformal group.

Abstract:We introduce simple group-theoretic techniques for classifying conformallyinvariant tensor structures. With them, we classify tensor structures of general n-point functions of non-conserved operators, and n ≥ 4-point functions of general conserved currents, with or without permutation symmetries, and in any spacetime dimension d. Our techniques are useful for bootstrap applications. The rules we derive simultaneously count tensor structures for flat-space scattering amplitudes in d + 1 dimensions.

We review some aspects of harmonic analysis for the Euclidean conformal group, including conformally-invariant pairings, the Plancherel measure, and the shadow transform. We introduce two efficient methods for computing these quantities: one based on weightshifting operators, and another based on Fourier space. As an application, we give a general formula for OPE coefficients in Mean Field Theory (MFT) for arbitrary spinning operators. We apply this formula to several examples, including MFT for fermions and "seed" operators in 4d, and MFT for currents and stress-tensors in 3d.Mean Field Theory (MFT) provides some of the simplest examples of crossing-symmetric, conformally-invariant correlation functions. Correlators in MFT are simply sums of products of two-point functions. In theories exhibiting large-N factorization, MFT is the leading contribution at large-N . For example, in AdS/CFT, MFT is the leading contribution to correlators in bulk perturbation theory [1][2][3]. In the analytic conformal bootstrap, MFT is the leading contribution to correlators at large spin [4][5][6][7][8][9][10]. MFT provides crucial example data for the numerical bootstrap [11], especially for spinning operators [12][13][14][15][16]. Furthermore, MFT OPE coefficients form the "ladder kernel" in SYK-like models [17][18][19][20][21][22][23]. Consequently, the OPE data of MFT (i.e. the scaling dimensions and OPE coefficients) is the starting point for many computations.Although correlators in MFT are simple, the OPE data can be nontrivial. OPE coefficients for a four-point function of fundamental scalars in MFT in 2-and 4-dimensions were guessed in [24]. 1 They were subsequently generalized to d-dimensions in [25] using a technique dubbed "conglomeration." In this work, we point out that conglomeration is part of a general toolkit of harmonic analysis for the Euclidean conformal group SO(d + 1, 1) [26]. Although harmonic analysis was first applied to CFTs in the 70's, it has played an especially important role in recent developments [17][18][19][20][21][22][23][27][28][29][30][31]. In section 2, we give an introduction to harmonic analysis for (Euclidean) CFTs.The calculation of [25] can be rephrased in terms of simple ingredients: the Plancherel measure, three-point pairings, and the "shadow transform" [32,33]. In particular, the computation of MFT OPE coefficients factorizes into two independent shadow transforms of threepoint functions, which are essentially generalizations of the famous "star-triangle" integral [34,35]. Using these observations, we write a general formula for OPE data of fundamental MFT fields in arbitrary Lorentz representations in section 2.8. Along the way, we derive orthogonality relations for conformal partial waves with arbitrary (internal and external) Lorentz representations.Our derivation essentially uses a "Euclidean inversion formula" -a formula that expresses OPE data as an integral of a four-point function over Euclidean space. MFT OPE data can also in principle be computed by applying the Lorentzian i...