Wronskian Solution for AdS/CFT Y-system

Using the discrete Hirota integrability we find the general solution of the full quantum Y-system for the spectrum of anomalous dimensions of operators in the planar AdS5/CFT4 correspondence in terms of Wronskian-like determinants parameterized by a finite number of Baxter’s Q-functions. We consider it as a useful step towards the construction of a finite system of non-linear integral equations (FiNLIE) for the full spectrum. The explicit asymptotic form of all the Q-functions for the large size operators is presented. We establish the symmetries and the analyticity properties of the asymptotic Q-functions and discuss their possible generalization to any finite size operators.

💡 Research Summary

The paper tackles one of the most challenging aspects of the planar AdS₅/CFT₄ correspondence: the exact determination of the full quantum spectrum of anomalous dimensions through the Y‑system. The Y‑system, originally formulated as an infinite set of coupled non‑linear functional equations (the Thermodynamic Bethe Ansatz), is notoriously difficult to solve because it involves infinitely many Y‑functions, intricate analyticity constraints, and a non‑trivial set of crossing and symmetry relations.
The authors begin by recasting the Y‑system into its equivalent Hirota form, the T‑system, which lives on a two‑dimensional integer lattice labeled by indices (a,s). In this formulation each T‑function obeys the discrete Hirota bilinear relation, a manifestation of the underlying integrable structure and fusion hierarchy. The crucial step is then to express every T‑function as a Wronskian determinant built from a finite set of Baxter Q‑functions. This Wronskian representation is a direct analogue of the classical construction for integrable spin chains, but here it is adapted to the full AdS/CFT spectral problem.
Specifically, the authors introduce eight fundamental Q‑functions (and their complex conjugates) together with a set of auxiliary “dual” Q‑functions. These functions depend on the spectral parameter u and are defined on a Riemann surface with a prescribed set of Zhukovsky cuts. The T‑functions are written as
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