Exact Spectrum of Anomalous Dimensions of Planar N = 4 Supersymmetric Yang–Mills Theory: TBA and excited states

Abstract

Using the thermodynamic Bethe ansatz method we derive an infinite set of integral non-linear equations for the spectrum of states/operators in AdS/CFT. The Y-system conjectured in [1] for the spectrum of all operators in planar N = 4 SYM theory follows from these equations. In particular, we present the integral TBA type equations for the spectrum of all operators within the sl(2) sector. We prove that all the kernels and free terms entering these TBA equations are real and have nice fusion properties in the relevant mirror kinematics. We find the analogue of DHM formula for the dressing kernel in the mirror kinematics.

Resurgence of the dressing phase for AdS5 × S5

“…7 The function c TS p,q (g), when expanded at strong coupling gives precisely (4.25), containing infinitely many exponentially suppressed, i.e. non-perturbative, terms of the form e −4πng .…”

“…Discφ p,q (t + 1) , 6 The integral (4.16) is well-defined for g ∈ C such that | sinh 2 (2πgx)| > 1 for |x| large enough. This leads to two disjoint domains of analyticity separated by the line (e iθ g) = 0.…”

“…This is a very lucky case which simplifies 7 With a slight abuse of notation we denote the median resummation (4.28) with the same symbol as its transseries representation (4.25) having in mind that they both uniquely define the one and the same analytic function.…”

Computing three-point functions for short operators

“…Based on the experience that integrability techniques usually start with a large-volume limit, this would probably require the adaption of a full thermodynamic Bethe ansatz [44][45][46][47][48] or quantum spectral curve [49] machinery.…”

A next-to-leading Lüscher formula

“…In particular, the study of the finite-volume corrections [2][3][4][5][6][7][8][9] for the anomalous dimensions/string energies spectrum of AdS 5 /CF T 4 , has culminated in the formulation of the so-called Thermodynamic Bethe Ansatz (TBA) equations and Y-system [10][11][12][13][14][15][16][17][18], which in principle govern the spectrum exactly at any order of the coupling constant and the volume parameter. Very recently, the TBA equations have been reduced first to few non-linear integral (so-called FiNLIE) equations [19] (see [17,18,[20][21][22][23][24][25][26][27] for some previous developments in that direction), then to an impressively simple set of Riemann-Hilbert equations [28].…”