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Exact S-matrices for AdS_3/CFT_2

Changrim Ahn, Diego Bombardelli

TL;DR

The paper constructs exact $S$-matrices for RR-flux AdS$_3$/CFT$_2$ in two backgrounds, ${\rm AdS}_3\times S^3\times T^4$ and ${\rm AdS}_3\times S^3\times S^3\times S^1$, by fixing two-particle scattering from residual $su(1|1)$ (or $su(1|1)\times su(1|1)$) symmetry and BES-based scalar factors. Employing an analytic Bethe Ansatz and the diagonalization of the $su(1|1)$ transfer matrix, the authors derive all-loop asymptotic Bethe Ansatz equations for both cases, with explicit expressions for transfer-matrix eigenvalues and magnonic variables. They demonstrate that their BAEs match the conjectured equations of Babichenko, Ohlsson Sax, and collaborators after suitable redefinitions of momenta and magnonic content, thereby supporting the proposed exact $S$-matrices. The work also discusses open issues, including the incomplete crossing structure for $su(1|1)$, the role of massless modes, and potential extensions to related AdS/CFT contexts, laying groundwork for deeper understanding of AdS$_3$/CFT$_2$ integrability.

Abstract

We propose exact $S$-matrices for the AdS_3/CFT_2 duality between Type IIB strings on AdS_3 x S^3 x M_4 with M_4=S^3 x S^1 or T^4 and the corresponding two-dimensional conformal field theories. We fix the complete two-particle S-matrices for both those cases of AdS_3/CFT_2, on the basis of the symmetries su(1|1) and su(1|1) x su(1|1), respectively preserved by their vacua. A crucial justification comes from the derivation of the all-loop Bethe ansatz matching exactly the recent conjecture proposed by [1] and [2].

Exact S-matrices for AdS_3/CFT_2

TL;DR

The paper constructs exact -matrices for RR-flux AdS/CFT in two backgrounds, and , by fixing two-particle scattering from residual (or ) symmetry and BES-based scalar factors. Employing an analytic Bethe Ansatz and the diagonalization of the transfer matrix, the authors derive all-loop asymptotic Bethe Ansatz equations for both cases, with explicit expressions for transfer-matrix eigenvalues and magnonic variables. They demonstrate that their BAEs match the conjectured equations of Babichenko, Ohlsson Sax, and collaborators after suitable redefinitions of momenta and magnonic content, thereby supporting the proposed exact -matrices. The work also discusses open issues, including the incomplete crossing structure for , the role of massless modes, and potential extensions to related AdS/CFT contexts, laying groundwork for deeper understanding of AdS/CFT integrability.

Abstract

We propose exact -matrices for the AdS_3/CFT_2 duality between Type IIB strings on AdS_3 x S^3 x M_4 with M_4=S^3 x S^1 or T^4 and the corresponding two-dimensional conformal field theories. We fix the complete two-particle S-matrices for both those cases of AdS_3/CFT_2, on the basis of the symmetries su(1|1) and su(1|1) x su(1|1), respectively preserved by their vacua. A crucial justification comes from the derivation of the all-loop Bethe ansatz matching exactly the recent conjecture proposed by [1] and [2].

Paper Structure

This paper contains 8 sections, 27 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Spectrum and $S$-matrix
  3. ${\rm AdS}_3\times S^3\times T^4$
  4. ${\rm AdS}_3\times S^3\times S^3\times S^1$
  5. Derivation of asymptotic Bethe ansatz equations
  6. Diagonalization of the transfer matrix
  7. Comparison to the Bethe equations of Babichenko:2009dkOhlssonSax:2011ms
  8. Discussion

Figures (4)

  • Figure 1: ${\rm AdS}_3\times S^3\times T^4$: two momentum-carrying nodes (black dots) are connected to two magnonic nodes (circle).
  • Figure 2: ${\rm AdS}_3\times S^3\times S^3\times S^1$: four momentum-carrying nodes (black dots) are connected to a single magnonic node (circle).
  • Figure 3: ${\rm AdS}_3\times S^3\times T^4$: two momentum-carrying nodes (black dots) are connected to four magnonic nodes (circle) after redefinition.
  • Figure 4: ${\rm AdS}_3\times S^3\times S^3\times S^1$: four momentum-carrying nodes (black dots) are connected to two magnonic nodes (circle) after redefinition.