Crossing symmetry, transcendentality and the Regge behaviour of 1d CFTs

Abstract

We develop the technology for Polyakov-Mellin (PM) bootstrap in onedimensional conformal field theories (CFT 1). By adding appropriate contact terms, we bootstrap various effective field theories in AdS 2 and analytically compute the CFT data to one loop. The computation can be extended to higher orders in perturbation theory, if we ignore mixing, for any external dimension. We develop PM bootstrap for O(N) theories and derive the necessary contact terms for such theories (which also involves a new higher gradient contact term absent for N = 1). We perform cross-checks which include considering the diagonal limit of the 2d Ising model in terms of the 1d PM blocks. As an independent check of the validity of the results obtained with PM bootstrap, we propose a suitable basis of transcendental functions, which allows to fix the four-point correlators of identical scalar primaries completely, up to a finite number of ambiguities related to the number of contact terms in the PM basis. We perform this analysis both at tree level (with and without exchanges) and at one loop. We also derive expressions for the corresponding CFT data in terms of harmonic sums. Finally, we consider the Regge limit of one-dimensional correlators and derive a precise connection between the latter and the large-twist limit of CFT data. Exploiting this result, we study the crossing equation in the three OPE limits and derive some universal constraints for the large-twist limit of CFT data in Regge-bounded theories with a finite number of exchanges.

Analytic bootstrap and Witten diagrams for the ABJM Wilson line as defect CFT1

“…Notably, the same approach led to analogous results in a less supersymmetric scenario [12]. More generally, line defects provide a useful and physically interesting laboratory for the application of the analytical techniques developed in the context of one-dimensional CFTs [13][14][15][16][17][18]. Finite-coupling results on defect conformal data were also obtained using integrability via the quantum spectral curve technology [19].…”

“…where is a small parameter, whose precise relation with the string tension cannot be predicted by symmetry considerations. Following [11,18], we start with the following Ansatz for the first order correction tof…”

“…One can of course choose C w (λ) ≡ 4B 1/2 (λ), so to realize a direct identification of w a with the O a in view of (3.53). 18 This would just correspond to an overall rescaling of the fields. 19 By evaluating perturbatively the two-point function (5.17) (namely, calcu- 18 This is the formal choice of [5], where the analogue relation is to the N = 4 SYM Bremsstrahlung function [10,82,83].…”

How to succeed at Witten diagram recursions without really trying

“…Exchange Witten diagrams are also intimately related to the method of analytic functional bootstrap [37-39, 45, 74-80] (see also [81][82][83][84]), as they neatly encapsulate the information of the functionals. More precisely, the Polyakov-Regge blocks can be decomposed into conformal blocks in two channels.…”

Loops in AdS: from the spectral representation to position space. Part II

“…Most of these computations are restricted to tree-level amplitudes. Other computations rely on the conformal symmetry or the supersymmetry of the dual theory [41][42][43][44][45][46][47][48][49][50][51][52][53][54]. For additional works on AdS amplitudes and in particular loop amplitudes, see [55][56][57][58][59][60][61][62][63][64][65][66][67][68][69][70][71].…”