Bootstrability in defect CFT: integrated correlators and sharper bounds

Abstract

We continue to develop Bootstrability — a method merging Integrability and Conformal Bootstrap to extract CFT data in integrable conformal gauge theories such as $$ \mathcal{N} $$ N = 4 SYM. In this paper, we consider the 1D defect CFT defined on a $$ \frac{1}{2} $$ 1 2 -BPS Wilson line in the theory, whose non-perturbative spectrum is governed by the Quantum Spectral Curve (QSC). In addition, we use that the deformed setup of a cusped Wilson line is also controlled by the QSC. In terms of the defect CFT, this translates into two nontrivial relations connecting integrated 4-point correlators to cusp spectral data, such as the Bremsstrahlung and Curvature functions — known analytically from the QSC. Combining these new constraints and the spectrum of the 10 lowest-lying states with the Numerical Conformal Bootstrap, we obtain very sharp rigorous numerical bounds for the structure constant of the first non-protected state, giving this observable with seven digits precision for the ’t Hooft coupling in the intermediate coupling region $$ \frac{\sqrt{\lambda }}{4\pi}\sim 1 $$ λ 4 π ~ 1 , with the error decreasing quickly at large ’t Hooft coupling. Furthermore, for the same structure constant we obtain a 4-loop analytic result at weak coupling. We also present results for excited states.

Ironing out the crease

“…In the former, it requires the 4-point function of displacements and in the latter the 2-point function of the singlet displacement. The 4-point function of the displacement operator in the Wilson loop is well studied [71][72][73][74] and nontrivial relations among them were found in [75,76]. The same was done for the surface operators in [5,76], but it certainly deserves further study.…”

Conformal dispersion relations for defects and boundaries

“…Some of these questions can be made very precise in the context of defect Conformal Field Theories (dCFT) [1], where this rich interplay between bulk and defect finds an explicit realization in the defect crossing equation. The latter is the central ingredient of the defect bootstrap program, an ambitious endeavour which has recently expanded in various interesting directions, such as the study of line and surface defects in holographic theories [2][3][4][5][6][7][8][9], the classification of boundaries and defects in free theories [10][11][12][13][14], the analysis of general boundaries in CFTs and the application to statistical systems [15][16][17][18][19][20][21][22][23][24][25], the bootstrability program aimed to an exact solution of a defect CFT [26,27] or the study of superconformal defects [28][29][30][31][32][33][34][35][36][37][38][39][40][41][42][43].…”

Bootstrapping $\mathcal{N}=4$ sYM correlators using integrability

“…It was recently proposed that integrability and bootstrap techniques could be combined to solve N = 4 sYM in the planar limit [30,31]. The authors introduced an approach ("Bootstrability") to study the 1D defect CFT defined by inserting local operators along a In this paper, we tackle a similar problem for a fully four-dimensional correlator involving four stress tensors in the planar limit of N = 4 sYM theory, meaning the N c → ∞ of a SU(N c ) gauge theory with fixed 't Hooft coupling λ = g 2 YM N c .…”

“…While our work is similar in spirit to refs. [30,31], the passage from D = 1 defects to D = 4 correlators presents significant challenges. A main one is that an infinite number of double-trace operators enter the OPE, polluting it with undesirable operators about which we have no spectral information.…”