Extended Y-system for the $AdS_5/CFT_4$ correspondence

TL;DR

The paper develops an extended Y-system for the AdS${}_5$/CFT${}_4$ spectral problem by incorporating local square-root discontinuity relations, capturing the intricate branch structure and the dressing factor. This local discontinuity data, together with the Y-system and asymptotic constraints, is shown to reproduce the ground-state Thermodynamic Bethe Ansatz equations for both the mirror and direct theories through a novel Cauchy-dispersion approach. The construction unifies the treatment across $Q$-, $y$-, $w$-, and $v$-particle sectors and clarifies the role of the dressing kernel via integral representations that relate mirror and direct dressings. The method opens a path toward excited-state TBA by extending the same discontinuity framework, and suggests deeper links between Y-/T-systems and potential underlying lattice structures. The results provide a conceptually transparent route to the finite-size spectrum in this integrable AdS/CFT context, with implications for exact non-perturbative checks and further analytic developments.

Abstract

We study the analytic properties of the $AdS_5/CFT_4$ Y functions. It is shown that the TBA equations, including the dressing factor, can be obtained from the Y-system with some additional information on the square-root discontinuities across semi-infinite segments in the complex plane. The Y-system extended by the discontinuity relations constitutes a fundamental set of local functional constraints that can be easily transformed into integral form through Cauchy's theorem.

Figures (7)

• Figure 1: The Y-system diagram corresponding to the $\text{AdS}_5/\text{CFT}_4$ TBA equations.
• Figure 2: The contour $\bar{\gamma}_{\sf o}$.
• Figure 3: The contour $\bar{\gamma}_{\sf x}$.
• Figure 4: The second sheet image $u_*$ of $u$.
• Figure 5: The deformed contour $\Gamma_{\sf O}$.