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Geometry of Massless Scattering in Integrable Superstring

Andrea Fontanella, Alessandro Torrielli

TL;DR

The paper studies massless non-relativistic scattering in AdS/CFT by embedding the massless R-matrix into a q-deformed Poincaré superalgebra, revealing a non-relativistic rapidity $\gamma$ that makes the R-matrix a function of $\gamma_1-\gamma_2$. It shows a $\theta \rightarrow \gamma$ substitution relating relativistic and non-relativistic massless S-matrices, validates this in AdS3 and AdS2 settings for the matrix part and, where possible, the dressing factor, and introduces a geometric, fibre-bundle framework to classify R-matrices via a connection $\Gamma$ and a constant matrix $\mathcal{A}$. The results unify the relativistic and non-relativistic pictures, provide explicit dressing-factor constructions in AdS3 and AdS2, and propose a general program to classify R-matrices by differential-geometric data, with applications to Bethe Ansatz and crossing symmetry.

Abstract

We consider the action of the $q$-deformed Poincaré superalgebra on the massless non-relativistic R-matrix in ordinary (undeformed) integrable $AdS_2 \times S^2 \times T^6$ type IIB superstring theory. The boost generator acts non-trivially on the R-matrix, confirming the existence of a non-relativistic rapidity $γ$ with respect to which the R-matrix must be of difference form. We conjecture that from a massless AdS/CFT integrable relativistic R-matrix one can obtain the parental massless non-relativistic R-matrix simply by replacing the relativistic rapidity with $γ$. We check our conjecture in ordinary (undeformed) $AdS_n \times S^n \times T^{10 - 2n}$, $n = 2, 3$. In the case $n=3$, we check that the matrix part and the dressing factor - up to numerical accuracy for real momenta - obey our prescription. In the $n=2$ case, we check the matrix part and propose the non-relativistic dressing factor. We then start a programme of classifying R-matrices in terms of connections on fibre bundles. The conditions obtained for the connection are tested on a set of known integrable R-matrices.

Geometry of Massless Scattering in Integrable Superstring

TL;DR

The paper studies massless non-relativistic scattering in AdS/CFT by embedding the massless R-matrix into a q-deformed Poincaré superalgebra, revealing a non-relativistic rapidity that makes the R-matrix a function of . It shows a substitution relating relativistic and non-relativistic massless S-matrices, validates this in AdS3 and AdS2 settings for the matrix part and, where possible, the dressing factor, and introduces a geometric, fibre-bundle framework to classify R-matrices via a connection and a constant matrix . The results unify the relativistic and non-relativistic pictures, provide explicit dressing-factor constructions in AdS3 and AdS2, and propose a general program to classify R-matrices by differential-geometric data, with applications to Bethe Ansatz and crossing symmetry.

Abstract

We consider the action of the -deformed Poincaré superalgebra on the massless non-relativistic R-matrix in ordinary (undeformed) integrable type IIB superstring theory. The boost generator acts non-trivially on the R-matrix, confirming the existence of a non-relativistic rapidity with respect to which the R-matrix must be of difference form. We conjecture that from a massless AdS/CFT integrable relativistic R-matrix one can obtain the parental massless non-relativistic R-matrix simply by replacing the relativistic rapidity with . We check our conjecture in ordinary (undeformed) , . In the case , we check that the matrix part and the dressing factor - up to numerical accuracy for real momenta - obey our prescription. In the case, we check the matrix part and propose the non-relativistic dressing factor. We then start a programme of classifying R-matrices in terms of connections on fibre bundles. The conditions obtained for the connection are tested on a set of known integrable R-matrices.

Paper Structure

This paper contains 20 sections, 2 theorems, 138 equations, 1 table.

Table of Contents

  1. Introduction
  2. The $q$-deformed Poincaré superalgebra
  3. $AdS_2$ and Bethe ansatz
  4. Plan of this paper
  5. $q$-deformed Poincaré for massless $AdS_2 \times S^2 \times T^6$
  6. $\gamma$ as a non-relativistic rapidity
  7. The dressing factor
  8. $AdS_3 \times S^3 \times T^4$
  9. $AdS_2 \times S^2 \times T^6$
  10. Universal R-matrix and Connections
  11. System of equations for $(\Gamma , \mathcal{A})$
  12. Physical unitarity
  13. Braiding unitarity
  14. Algebra invariance
  15. Conclusions
  16. ...and 5 more sections

Key Result

Theorem 1

Continuous deformations of the solution $\mathcal{A} = \Pi$ are not solutions.

Theorems & Definitions (5)

  • Conjecture 1
  • Theorem 1
  • proof
  • Theorem 2
  • proof