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$AdS_5\times S^5$ mirror TBA equations from Y-system and discontinuity relations

Janos Balog, Arpad Hegedus

Abstract

Using the recently proposed set of discontinuity relations we translate the AdS/CFT Y-system to TBA integral equations and quantization conditions for a large subset of excited states from the sl(2) sector of the $AdS_5 \times S^5$ string sigma-model. Our derivation provides an analytic proof of the fact that the exact Bethe equations reduce to the Beisert-Staudacher equations in the asymptotic limit. We also construct the corresponding T-system and show that in the language of T-functions the energy formula reduces to a single term which depends on a single T-function.

$AdS_5\times S^5$ mirror TBA equations from Y-system and discontinuity relations

Abstract

Using the recently proposed set of discontinuity relations we translate the AdS/CFT Y-system to TBA integral equations and quantization conditions for a large subset of excited states from the sl(2) sector of the string sigma-model. Our derivation provides an analytic proof of the fact that the exact Bethe equations reduce to the Beisert-Staudacher equations in the asymptotic limit. We also construct the corresponding T-system and show that in the language of T-functions the energy formula reduces to a single term which depends on a single T-function.

Paper Structure

This paper contains 42 sections, 343 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Starting relations and assumptions
  3. TBA equations with cuts
  4. TBA lemma 1a
  5. TBA lemma 1b
  6. TBA lemma 2
  7. TBA integral equations
  8. Construction of the T-system
  9. Chain lemma
  10. Constructing $T_{a,0}$
  11. Constructing $T_{a,1}$
  12. The complete T-system
  13. Gauge transformations
  14. Large $\vert u\vert$ asymptotics
  15. Dispersion relation for $Y_-$
  16. ...and 27 more sections

Figures (4)

  • Figure 1: The ${\rm AdS}_5{\rm /CFT}_4$ Y-system. Full circles on the $a$ axis correspond to massive nodes with $s=0$. The $s$ axis corresponds to nodes with $a=1$.
  • Figure 2: The ${\rm AdS}_5{\rm /CFT}_4$ T-system. The $s$ axis corresponds to nodes with $a=0$.
  • Figure 3: The contour $\Gamma_0$. It goes around all positive and negative even cuts.
  • Figure 4: This contour first goes parallel to the real axis and goes around the cuts at $Z+2m$, $Z$ positive integer, $m=0,1,\dots$