Light-ray operators in conformal field theory

Abstract

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. ...

Unitarity methods in AdS/CFT

“…First, by manipulating AdS bulk-to-bulk propagators, one can recast Witten diagrams as spectral integrals over conformally-invariant gluings of CFT three-point functions. Second, one employs techniques from harmonic analysis to perform crossing transformations [40,41] and position-space integrals [42][43][44] of these conformal structures. In this way, any n-point, loop-level diagram can be expressed in a conformal partial wave (CPW) expansion.…”

“…By working with operators on the principal series, we will leverage this equality when 9 We also adopt the convention that inside two and three-point functions O denotes an abstract operator with dimension ∆ O and does not include any factors from the shadow transform. 10 In Euclidean signature the integration is over all of space, but in Lorentzian signature we can consider more general pairings of operators and domains of integration [42,56]. 11 Note that the 6j symbol is known explicitly in closed form only in d = 1, 2, 4 [27,57].…”

“…The computation follows [27] and employs the technical toolkit of conformal integrals reviewed in Section 2 (see e.g. [27,[41][42][43]60]).…”

“…The definition of the three-point pairing [42,43] is given in (A.15). This gives the same CPW decomposition as before, but written in a different way: found in [68] by brute force calculation using the methods of [24].…”

The gravity dual of Lorentzian OPE blocks

“…• In this work, we have specifically constructed the holographic description of the OPE block by considering the shadow projector. However in the Lorentzian CFT's, it is known that we can also have spin-shadow and light ray transformations [33,38], such that the full projector should include these transformations as they are also part of the D 8 Weyl group of the Lorentzian conformal symmetry group SO (2, d). It would be very interesting to generalize our construction to consider the OPE blocks involving these non-local exchange operators, and how causality conditions can modify the definitions of HKLL-type representation of their bulk dual fields.…”