Review of AdS/CFT Integrability, Chapter III.7: Hirota Dynamics for Quantum Integrability

Abstract

We review recent applications of the integrable discrete Hirota dynamics (Y-system) in the context of calculation of the planar AdS/CFT spectrum. We start from the description of solution of Hirota equations by the Bäcklund method where the requirement of analyticity results in the nested Bethe ansatz equations. Then we discuss applications of the Hirota dynamics for the analysis of the asymptotic limit of long operators in the AdS/CFT Y-system.

Solving the AdS/CFT Y-system

“…for an important case of the principal chiral field [4,6]): even the analyticity properties of T-functions appear to be considerably constrained by the structure of Hirota equation. All these observations should hold for the AdS/CFT Y-systems [7], and in particular for the most studied AdS 5 /CFT 4 case, solving the problem of planar spectrum of anomalous dimensions in N=4 SYM theory (see for the recent review [8], and in particular the chapters [9,10] therein). The AdS 5 /CFT 4 Y-and T-systems with T-hook boundary conditions proposed in [7] and summarized in figure 1 were later shown to be equivalent, with certain analyticity requirements [11][12][13], to the TBA equations [14][15][16].…”

“…For the detailed description of the Y-system for AdS/CFT we refer to [8][9][10]. In this section we list the main results and general facts about Y-systems and Hirota equation.…”

“…2 In addition to the shape of the pa, sq domain one should also specify the analyticity properties of the Y-functions. These properties can be deduced from the TBA equations [14][15][16] and were explicitly written in [9,11,13].…”

“…The analyticity properties of Y-functions which are expected to be satisfied by the physical solutions of the Y-system [9] are written in the table 1, in the notations of the previous section. These Y-functions are considered on the mirror sheet which means that we have chosen the long Z-cuts [16].…”

“…A natural assumption, which will be later demonstrated, is that, in a certain gauge which we denote by the bold font, T a,s has good analyticity properties for the upper band, whether as in another gauge, denoted by the "blackboard" font, T a,s has good analyticity properties for the right band of the T-hook. 9 These analyticity properties are explicitly written in table 2.…”