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Wronskian Solution for AdS/CFT Y-system

Nikolay Gromov, Vladimir Kazakov, Sebastien Leurent, Zengo Tsuboi

TL;DR

This work derives an explicit Wronskian determinant representation for the AdS$_5$/CFT$_4$ Y-system in the ${\mathbb T}$-hook, expressing the $T$- and $Y$-functions through a finite set of Baxter ${\mathsf Q}$-functions and a generating functional. By establishing QQ-relations and a complete set of determinant formulas, it reduces the problem to eight basis ${\mathsf Q}$-functions, paving the way for a finite-size nonlinear integral equation (FiNLIE) description of the spectrum. The construction connects the classical finite-gap / monodromy picture (via $SU(2,2|4)$ characters) to the quantum Y-system, and clarifies symmetry and analyticity properties, including mirror reality and wing-exchange. The results offer a concrete framework toward FiNLIE for the full planar AdS$_5$/CFT$_4$ spectrum and suggest extensions to operatorial Q-operators and related dualities such as ABJM.

Abstract

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.

Wronskian Solution for AdS/CFT Y-system

TL;DR

This work derives an explicit Wronskian determinant representation for the AdS/CFT Y-system in the -hook, expressing the - and -functions through a finite set of Baxter -functions and a generating functional. By establishing QQ-relations and a complete set of determinant formulas, it reduces the problem to eight basis -functions, paving the way for a finite-size nonlinear integral equation (FiNLIE) description of the spectrum. The construction connects the classical finite-gap / monodromy picture (via characters) to the quantum Y-system, and clarifies symmetry and analyticity properties, including mirror reality and wing-exchange. The results offer a concrete framework toward FiNLIE for the full planar AdS/CFT spectrum and suggest extensions to operatorial Q-operators and related dualities such as ABJM.

Abstract

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.

Paper Structure

This paper contains 22 sections, 62 equations, 4 figures.

Table of Contents

  1. Introduction
  2. $Y$-system and $T$-system for the spectrum of AdS/CFT
  3. $SU(2,2|4)$ character solution of the Y-system and classical limit
  4. The character solution of the simplified AdS/CFT Y-system in $\mathbb{T}$-hook
  5. Quantization of the classical algebraic curve
  6. Solution of the full Hirota equation in $\mathbb{T}$-hook
  7. Generating functional for the $\mathbb{T}$-hook
  8. QQ-relations
  9. Explicit Wronskian formulae for $\mathbb{T}$-hook
  10. A basis for ${\mathsf Q}$ functions
  11. Asymptotic expressions for the generating functional and ${\mathsf Q}$ functions
  12. Asymptotic limit for the generating functional
  13. Explicit expressions for the asymptotic ${\mathsf Q}$-functions
  14. Physical symmetries
  15. Lower boundary:
  16. ...and 7 more sections

Figures (4)

  • Figure 1: Numerical results of solution of the AdS/CFT Y-system (in integral, TBA form) for Konishi dimension as a function of 't Hooft coupling $\lambda$ (from Gromov:2009zb and Frolov:2010wt).
  • Figure 2: T-shaped "fat hook" ($\mathbb{T}$-hook) uniting two ${\rm SU}(2|2)$ fat hooks, see Gromov:2009tv for this $\mathbb{T}$-hook and its generalization Hegedus:2009ky.
  • Figure 3: One can quantize the classical curve by choosing different orders of the sheets. Once an order (called path) is fixed the ${\mathsf Q}$ functions will correspond to the cuts crossing the line between two neighbor sheets. A ${\mathsf Q}$ function is not affected by the permutations of sheets above and below the line. Thus the ${\mathsf Q}$ function is uniquely determined by the subset of the sheets above the line and we denote it as $Q_{{\hat{1}}{{1}}{{2}}{\hat{2}}{\hat{3}}}$. For more formal definitions see section 4.3.
  • Figure 4: Hasse diagram for $gl(2|2)$ (cf. Tsuboi:2009ud): To construct the generating functional \ref{['eq:GenFn22']} we move along any path starting at the node $\emptyset$ and ending at the node ${{1}}{{2}}{\hat{1}}{\hat{2}}$. Each line corresponds to adding the next $U$-factor (or $V$-factor in \ref{['eq:GenFn224']} for the $(2,2|4)$ case) with one extra index. The dashed blue (resp. dotted red) lines stand for adding a "bosonic" (resp. fermionic) index, and each local change of the nesting path at some rectangular facet of the diagram gives rise to the $QQ$-relations \ref{['eq:QQb']}, \ref{['eq:QQf']}. The equation \ref{['eq:QQf']} (resp. \ref{['eq:QQb']}) corresponds to the facets having $2$ "bosonic" lines and $2$ "fermionic" lines (resp to the facets having all $4$ "bosonic" or all $4$ "fermionic" lines).