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The classical R-matrix of AdS/CFT and its Lie dialgebra structure

Benoit Vicedo

Abstract

The classical integrable structure of Z_4-graded supercoset sigma-models, arising in the AdS/CFT correspondence, is formulated within the R-matrix approach. The central object in this construction is the standard R-matrix of the Z_4-twisted loop algebra. However, in order to correctly describe the Lax matrix within this formalism, the standard inner product on this twisted loop algebra requires a further twist induced by the Zhukovsky map, which also plays a key role in the AdS/CFT correspondence. The non-ultralocality of the sigma-model can be understood as stemming from this latter twist since it leads to a non skew-symmetric R-matrix.

The classical R-matrix of AdS/CFT and its Lie dialgebra structure

Abstract

The classical integrable structure of Z_4-graded supercoset sigma-models, arising in the AdS/CFT correspondence, is formulated within the R-matrix approach. The central object in this construction is the standard R-matrix of the Z_4-twisted loop algebra. However, in order to correctly describe the Lax matrix within this formalism, the standard inner product on this twisted loop algebra requires a further twist induced by the Zhukovsky map, which also plays a key role in the AdS/CFT correspondence. The non-ultralocality of the sigma-model can be understood as stemming from this latter twist since it leads to a non skew-symmetric R-matrix.

Paper Structure

This paper contains 36 sections, 6 theorems, 90 equations, 1 figure.

Table of Contents

  1. Introduction and summary
  2. Remarks.
  3. $\mathbb{Z}_4$-graded supercoset $\sigma$-models
  4. Lie superalgebra.
  5. Supercoset $\sigma$-model.
  6. Hamiltonian analysis of GS.
  7. Hamiltonian analysis of PS.
  8. Lax connection.
  9. $r/s$-matrix algebra.
  10. Twisted loop algebra with twisted inner product
  11. Loop algebra.
  12. Let's do the twist.
  13. Come on let's twist again.
  14. Smooth dual.
  15. Lax matrix.
  16. ...and 21 more sections

Key Result

Lemma 3.1

For any $b = \sum_{i = 0}^3 b^{(i)} \in \mathfrak{g}$ we have, formally,

Figures (1)

  • Figure 1: The black dots denote the poles of the projection kernels \ref{['proj kern resum twist']} at $i^n z_1$ in the $z_2$-plane. Figure $(a)$ shows the (red) contour $\Gamma_+$ for $\pi_+$ which contains all four singularities and the (blue) contour $\Gamma_-$ for $\pi_-$ which goes around the origin but contains none of the singularities. Figure $(b)$ shows the contour for the $R$-matrix, namely $\Gamma = \{ |z_2| = |z_1| \}$ with all four singularities cut out.

Theorems & Definitions (10)

  • Lemma 3.1
  • Lemma 7.1
  • proof
  • Theorem 7.2
  • proof
  • Lemma 7.3
  • proof
  • Proposition 7.4
  • proof
  • Proposition 7.5