Y-system and beta-deformed N=4 Super-Yang-Mills

TL;DR

The paper shows that perturbative anomalous dimensions in the real-$\beta$-deformed ${\cal N}=4$ SYM can be derived from the AdS$_5$/CFT$_4$ Y-system by introducing a twisted generating functional with four twists encoding the deformation. It develops a general asymptotic solution at large operator length $L$, decouples the Y-system into left/right wings, and analyzes the weak-coupling expansion to obtain wrapping corrections, reproducing known results such as the Konishi case and providing a generating function that directly yields perturbative integrals $I_j^{(L)}$. The approach is validated against the Beisert-Roiban ABA and demonstrates how the deformation parameter $\beta$ clarifies the connection between perturbative integrability and the Y-system. The work suggests a pathway to derive the Y-system from perturbation theory and motivates exploration of broader deformations. Overall, it strengthens the link between integrability-based methods and perturbative gauge theory in β-deformed settings, with practical means to extract higher-loop contributions.

Abstract

We show how the perturbation theory results recently obtained by F.Fiamberti, A.Santambrogio, C.Sieg and D.Zanon for operator anomalous dimensions of beta-deformed Super-Yang-Mills theory can be reproduced from the AdS5/CFT4 Y-system proposed by N.G., V.Kazakov and P.Vieira. To do this, we obtain the general twisted asymptotic solution of this Y-system of functional equations. We show that existence of an additional parameter beta in the deformed theory allows to extract rich information about the perturbation theory integrals directly from Y-system. Using this method we found a simple generating function for a broad class of such integrals.