# Geometry of Massless Scattering in Integrable Superstring

## TL;DR

This paper explores the structure of massless scattering matrices in integrable superstring theories, revealing a non-relativistic rapidity parameter and proposing a method to derive non-relativistic R-matrices from relativistic ones, with verification across different models.

## Contribution

It introduces a conjecture relating relativistic and non-relativistic R-matrices via a rapidity substitution and tests this in various AdS backgrounds, advancing understanding of integrable scattering.

## Key findings

- Confirmed the existence of a non-relativistic rapidity $\\gamma$ for the R-matrix.
- Validated the conjecture in $AdS_3$ and $AdS_2$ models, matching matrix parts and dressing factors.
- Proposed a classification approach for R-matrices using connections on fiber bundles.

## Abstract

We consider the action of the $q$-deformed Poincar\'e 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 $\gamma$ 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 $\gamma$. 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.