Solving the AdS/CFT Y-system

TL;DR

This work develops a finite, nonlinear integral equation framework (FiNLIE) for the planar AdS$_5$/CFT$_4$ spectrum by exploiting integrability, analyticity, and a quantum $ ext{Z}_4$ symmetry. Central to the construction is a Wronskian representation of T-functions in terms of a finite set of Q-functions, together with a careful choice of a magic sheet that reduces the Y/T-system’s analytic complexity. The authors derive a closed, self-consistent set of FiNLIE equations that reproduce known Konishi-energy results from the original infinite Y-system and establish equivalence with the TBA. This approach sharpens the analytic control over the spectral problem and provides a practical route to precise weak- and potentially strong-coupling computations, with broad implications for understanding AdS/CFT integrability and related quantum field theories. The framework also highlights a path toward a quantum spectral-curve formulation based on a finite yet richly structured set of Q- and T-functions.

Abstract

Using integrability and analyticity properties of the AdS5/CFT4 Y-system we reduce it to a finite set of nonlinear integral equations. The Z4 symmetry of the underlying coset sigma model, in its quantum version, allows for a deeper insight into the analyticity structure of the underlying 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.

Figures (11)

• Figure 1: Boundary conditions for Y-system (b) and T-system (a) of AdS$_5$/CFT$_4$:$\mathbb{T}$-shaped "fat hook" ($\mathbb{T}$-hook). We will often distinguish in the ${\mathbb{T}}$-hook (a) the upper band -- all the nodes with $a\ge|s|$, the right band -- all the nodes with $a\le s$ and the left band -- all the nodes with $a\le|s|, s<0$.
• Figure 2: Riemann sheets of the multivalued function $x(u)$. The subfigure \ref{['fig:Xriemannl']} represents the function $u\mapsto x(u)$ (upper sheet) and $u\mapsto 1/x(u)$ (lower sheet), whereas the subfigure \ref{['fig:Xriemanns']} represents the function $u\mapsto \hat{x}(u)$ (upper sheet) and $u\mapsto 1/\hat{x}(u)$ (lower sheet). When $u$ is real, $x^{[\pm 1]}$ is defined by an analytic continuation from the real axis, along the red path which avoids the cut ${\hbox{$\space\check {\space\hbox{$Z$}}$}}_0$, whereas $\hat{x}^{[\pm 1]}$ is defined by a continuation along the green path which avoids the cut ${\widehat{ Z}}_0$. We see that $x^{[+1]}= \hat{x}^{[+1]}$ whereas $x^{[-1]} = 1/ \hat{x}^{[-1]}$.
• Figure 3: Comparison of the mirror (on the left) and the magic sheets (on the right).${\cal T}_{1,s}$ coincide on the real axis of both sheets.
• Figure 4: Paths used for analytical continuation. To get the Bethe equations used in the TBA approach, we continue $Y_{1,0}$ using the path $\gamma$ (on the left). Alternatively we can formulate Bethe equations using ${\mathbb{T}}_{1,1}$ continued over $\gamma_+$ and $\gamma_-$ (on the right).
• Figure 5: Infinite band $B^{(\mathfrak{n})}$ for the Hirota equation.