The analytic bootstrap in fermionic CFTs

110

158

We initiate an approach to constraining conformal field theory (CFT) data at finite temperature using methods inspired by the conformal bootstrap for vacuum correlation functions. We focus on thermal one-and two-point functions of local operators on the plane. The KMS condition for thermal two-point functions is cast as a crossing equation. By studying the analyticity properties of thermal two-point functions, we derive a "thermal inversion formula" whose output is the set of thermal one-point functions for all operators appearing in a given OPE. This involves identifying a kinematic regime which is the analog of the Regge regime for four-point functions. We demonstrate the effectiveness of the inversion formula by recovering the spectrum and thermal one-point functions in mean field theory, and computing thermal onepoint functions for all higher-spin currents in the critical O(N ) model at leading order in 1/N . Furthermore, we develop a systematic perturbation theory for thermal data in the large spin, low-twist spectrum of any CFT. We explain how the inversion formula and KMS condition may be combined to algorithmically constrain CFTs at finite temperature. Throughout, we draw analogies to the bootstrap for vacuum four-point functions. Finally, we discuss future directions for the thermal conformal bootstrap program, emphasizing applications to various types of CFTs, including those with holographic duals.

Section: Discussionmentioning

confidence: 83%

“…Combining the left-and right-moving blocks yields the full scaling behavior of the torus one-point blocks at high temperature. 45 It is remarkable that, for…”

Section: (A7)mentioning

confidence: 99%

See 1 more Smart Citation

The conformal bootstrap at finite temperature

Iliesiu

Kologlu

Mahajan

et al. 2018

110

158

“…One concrete result of this work is a generalization of Caron-Huot's Lorentzian inversion formula to four-point correlators of operators in arbitrary Lorentz representations. Caron-Huot's original formula has already proven useful in a variety of contexts [89][90][91][92][93][94][95], 67 and we hope that our generalization will be similarly useful. For example, one might try to determine all four-point functions in theories with weakly-broken higher spin symmetry, generalizing the results of [93].…”

Section: Discussionmentioning

confidence: 85%

Light-ray operators in conformal field theory

Kravchuk

Simmons-Duffin

2018

268

592

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. ...

“…Another interesting problem is the extension of the present methods to other CFTs. Large spin perturbation theory has been successfully applied to several models at leading order, including cubic models in six dimensions and large-N critical models [9], weakly coupled gauge theories [9,35] and fermionic CFTs [36,37]. Another interesting family of CFTs are the multicritical models, studied e.g.…”

Section: Discussionmentioning

confidence: 99%

Taming the ϵ-expansion with large spin perturbation theory

Alday

Henriksson

Loon

2018

Self Cite

159

We apply analytic bootstrap techniques to the four-point correlator of fundamental fields in the Wilson-Fisher model. In an -expansion crossing symmetry fixes the double discontinuity of the correlator in terms of CFT data at lower orders. Large spin perturbation theory, or equivalently the recently proposed Froissart-Gribov inversion integral, then allows one to reconstruct the CFT data of intermediate operators of any spin. We use this method to compute the anomalous dimensions and OPE coefficients of leading twist operators. To cubic order in the double discontinuity arises solely from the identity operator and the scalar bilinear operator, making the computation straightforward. At higher orders the double discontinuity receives contributions from infinite towers of higher spin operators. At fourth order, the structure of perturbation theory leads to a proposal in terms of functions of certain degree of transcendentality, which can then be fixed by symmetries. This leads to the full determination of the CFT data for leading twist operators to fourth order.