Solving the AdS/CFT Y-system

Abstract

Using integrability and analyticity properties of the AdS 5 /CFT 4 Y-system we reduce it to a finite set of nonlinear integral equations. The Z 4 symmetry of the underlying coset sigma model, in its quantum version, allows for a deeper insight into the analyticity structure of the corresponding Y-functions and T-functions, as well as for their analyticity friendly parameterization in terms of Wronskian determinants of Q-functions. As a check for the new equations, we reproduce the numerical results for the Konishi operator previously obtained from the original infinite Y-system.

Quantum spectral curve for arbitrary state/operator in AdS5/CFT4

“…The Y-system together with these analyticity relations can be called analytic Y-system. It was noticed in [12] that these relations can be further simplified by introducing T-functions related to Y-functions as follows which is the so-called discrete Hirota equation. But most importantly, the discontinuity relations of [9] turn out to be very simple statements about the analyticity of T-functions.…”

“…But most importantly, the discontinuity relations of [9] turn out to be very simple statements about the analyticity of T-functions. In particular, as was shown in [12], (3.2) is equivalent to a statement of existence of a special gauge choice for T-functions, denoted as T in [12], such that T a−1,0 , T a,±1 , T a+1,±2 have no cuts inside the strip |Im u| < a 2 (3.5)…”

“…The integrability of the latter would imply that it can be rewritten in terms of a finite number of Q-functions of the spectral parameter -analogues of the Baxter polynomials in the studies of integrable spin chains [10,11]. This venue was explored in our paper [12] where, using this discrete classical integrability and an important analyticity input, a finite system of non-linear integral equations (FiNLIE), somewhat reminding in spirit the Destri-De Vega equations, was formulated and successfully tested numerically.…”

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].…”