q-Deformation of the AdS5 x S5 Superstring S-matrix and its Relativistic Limit

TL;DR

This paper constructs and analyzes a q-deformed S-matrix for the AdS5 x S5 superstring, exploring its properties, limits, and relation to relativistic theories and integrable models.

Contribution

It introduces a new family of q-deformed, factorizable S-matrices for superstring excitations, including their scalar factors and relativistic limits.

Findings

The S-matrices satisfy unitarity and crossing equations.

In the relativistic limit, the mirror S-matrix aligns with semi-symmetric space sine-Gordon theory.

The quantum deformation interpolates between different classical limits of the integrable system.

Abstract

A set of four factorizable non-relativistic S-matrices for a multiplet of fundamental particles are defined based on the R-matrix of the quantum group deformation of the centrally extended superalgebra su(2|2). The S-matrices are a function of two independent couplings g and q=exp(i\pi/k). The main result is to find the scalar factor, or dressing phase, which ensures that the unitarity and crossing equations are satisfied. For generic (g,k), the S-matrices are branched functions on a product of rapidity tori. In the limit k->infinity, one of them is identified with the S-matrix describing the magnon excitations on the string world sheet in AdS5 x S5, while another is the mirror S-matrix that is needed for the TBA. In the g->infinity limit, the rapidity torus degenerates, the branch points disappear and the S-matrices become meromorphic functions, as required by relativistic S-matrix theory. However, it is only the mirror S-matrix which satisfies the correct relativistic crossing equation. The mirror S-matrix in the relativistic limit is then closely related to that of the semi-symmetric space sine-Gordon theory obtained from the string theory by the Pohlmeyer reduction, but has anti-symmetric rather than symmetric bound states. The interpolating S-matrix realizes at the quantum level the fact that at the classical level the two theories correspond to different limits of a one-parameter family of symplectic structures of the same integrable system.