Discrete Hirota dynamics for AdS/CFT

TL;DR

The paper develops a discrete Hirota framework for AdS/CFT by solving the Y- and T-systems on a T-shaped fat-hook via a chain of auto-Bäcklund transformations. This yields TT-, TQ-, and QQ-relations that express the infinite set of T-functions in terms of a finite set of boundary Q-functions, enabling a constructive path toward an NLIE description of the exact planar spectrum. The AdS/CFT case is worked out in detail for the (2,2) fat-hook, with explicit generating-series constructions and BR-formula-based reconstruction of all T-functions from nine boundary functions. The results provide a systematic, scalable approach to encode and potentially compute wrapping-corrected anomalous dimensions within a unified integrable framework.

Abstract

Recently a set of functional equations defining the anomalous dimensions of arbitrary local single trace operators in planar N=4 supersymmetric Yang-Mills theory has been conjectured. These functional equations take the form of a Y-system defined on a special shaped domain. This Y-system can be equivalently reformulated as a T-system defined on a "T-shaped fat hook". The elements of the T-system satisfy discrete Hirota equations. In the present paper the discrete Hirota equations for AdS/CFT are solved by means of a chain of Backlund transformations and as a result TT-, TQ-, and QQ-relations are obtained for AdS/CFT.

Figures (7)

• Figure 1: General T-shaped fat hook .
• Figure 2: Schematic representation of the actions of (BT1), (BT2), ($\overline{BT1}$) and ($\overline{BT1}$) Bäcklund transformations.
• Figure 3: Domain of definition for T-functions $T_{k,m}^{k',m'}(a,s,u)$.
• Figure 4: Degenerate domain corresponding to the choice $m=-m'$.
• Figure 5: Undressing paths for a T-shaped fat-hook of type $(K',M')\odot (K,M)$.