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PSU(2,2|4) Character of Quasiclassical AdS/CFT

Nikolay Gromov, Vladimir Kazakov, Zengo Tsuboi

TL;DR

This work solves the AdS/CFT Y-system in the strong coupling scaling limit by reducing the Hirota-based T-system to a Q-system that is captured by ${ m U}(2,2|4)$ super-characters. The authors derive explicit determinant formulae for the T-functions (T-hook) and interpret them as traces over infinite-dimensional unitary representations, linking them to the classical worldsheet monodromy matrix. They demonstrate a precise match between the asymptotic/Y-system solution and the quasiclassical one-loop spectrum of string theory, including wrapping effects, via the algebraic curve. The results highlight a deep connection between Y-system dynamics and the monodromy data, and point toward a quantum generalization with a finite set of nonlinear equations for the full spectrum at arbitrary coupling. This framework offers new avenues for exact spectral computations in AdS/CFT and unifies representation theory with integrable string dynamics at strong coupling.

Abstract

We solve the recently proposed T- and Y-systems (Hirota equation) for the exact spectrum of AdS/CFT in the strong coupling scaling limit for an arbitrary quasiclassical string state. The corresponding T-functions appear to be super-characters of the SU(2,2|4) group in unitary representations with a highest weight, with the classical AdS5xS5 superstring monodromy matrix as the group element. We propose a concise first Weyl-type formula for these characters and show that they correctly reproduce the results of quasiclassical one-loop quantization in all sectors of the superstring, under some natural assumptions. We also speculate about possible relation between the T-functions and the quantum monodromy matrix.

PSU(2,2|4) Character of Quasiclassical AdS/CFT

TL;DR

This work solves the AdS/CFT Y-system in the strong coupling scaling limit by reducing the Hirota-based T-system to a Q-system that is captured by super-characters. The authors derive explicit determinant formulae for the T-functions (T-hook) and interpret them as traces over infinite-dimensional unitary representations, linking them to the classical worldsheet monodromy matrix. They demonstrate a precise match between the asymptotic/Y-system solution and the quasiclassical one-loop spectrum of string theory, including wrapping effects, via the algebraic curve. The results highlight a deep connection between Y-system dynamics and the monodromy data, and point toward a quantum generalization with a finite set of nonlinear equations for the full spectrum at arbitrary coupling. This framework offers new avenues for exact spectral computations in AdS/CFT and unifies representation theory with integrable string dynamics at strong coupling.

Abstract

We solve the recently proposed T- and Y-systems (Hirota equation) for the exact spectrum of AdS/CFT in the strong coupling scaling limit for an arbitrary quasiclassical string state. The corresponding T-functions appear to be super-characters of the SU(2,2|4) group in unitary representations with a highest weight, with the classical AdS5xS5 superstring monodromy matrix as the group element. We propose a concise first Weyl-type formula for these characters and show that they correctly reproduce the results of quasiclassical one-loop quantization in all sectors of the superstring, under some natural assumptions. We also speculate about possible relation between the T-functions and the quantum monodromy matrix.

Paper Structure

This paper contains 24 sections, 130 equations, 6 figures.

Table of Contents

  1. Introduction
  2. Y-system of AdS/CFT and the characters of ${\rm U}(2,2|4)$
  3. Y-system for AdS/CFT: equations and definitions
  4. The Q-system limit and its ${\rm U}(2,2|4)$ "character" solution
  5. New determinant formulae
  6. Description of representations of ${\rm U}(2,2|4)$ for the AdS/CFT Y-system
  7. Description of the general construction
  8. Unitarity
  9. Comparing highest weight with the equation for characters
  10. Example: representations of ${\rm GL}(2)$
  11. Quantum $GL(1+1)$
  12. Solution of Y-system in the scaling limit and quasiclassical strings
  13. Asymptotic solution in scaling limit
  14. Asymptotic T-functions for the entire T-hook
  15. Classical limit
  16. ...and 9 more sections

Figures (6)

  • Figure 1: (Left: Fig.1a; Right: Fig.1b) T-shaped "fat hook" (T-hook) uniting two ${\rm SU}(2|2)$ fat hooks, see Gromov:2009tv for this T-hook and its generalization Hegedus:2009ky.
  • Figure 2: "Fat hook" where the representations of a $SU(M|N)$ symmetric super-spin chain live. See Tsuboi:1997iqKazakov:2007fy for the details on fat hooks and $T$-functions for spin chains related to superalgebras.
  • Figure 3: Dynkin diagram for the Lie superalgebra $\frak{gl}(4|4)$ with the Dynkin indexes corresponding to the characters in (2.18).
  • Figure 4: "T- hook"and the highest weight components arranged as a generalized Young diagram living inside the T-hook
  • Figure 5: Transformation property of the Dynkin labels under the fermionic duality. The duality transform the diagram in one grading to another. The dotted lines correspond to the fermionic grading whereas the solid lines represent bosonic grading.
  • ...and 1 more figures