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

TL;DR

The paper constructs a q-deformed, non-relativistic S-matrix for the AdS5 x S5 magnon system dependent on two couplings (g and q), and derives the scalar dressing phase required for unitarity and crossing. By analyzing the dispersion relation on a rapidity torus and exploring the string and relativistic limits, the authors establish a precise correspondence between magnon, mirror, and relativistic theories, and they show that only the mirror S-matrix remains consistent with relativistic crossing in the g 9fty limit. The dressing phase is obtained from a q-deformed Riemann-Hilbert problem, reducing to known magnon results in the appropriate limit and yielding explicit expressions for both r and its alternative rhat. Bound-state analyses reveal how atypical short representations organize the spectrum and how crossing ties the s- and t-channel poles across different sheets, with finite-k root-of-unity truncation offering a natural regularization candidate. Together, these results link the interpolating non-relativistic theory to relativistic integrable structures and point toward a consistent TBA framework for the interpolating mirror theory.

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(iπ/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.

Figures (7)

• Figure 1: The rapidity torus realized as a 4-fold cover of the cylinder parameterized by $q^{-2iu}$. The top and bottom cut are identified.
• Figure 2: Crossing symmetry of the basic 2-body S-matrix. Note that the right-hand diagram is just a $\frac{\pi}{2}$ rotation of the left-hand diagram.
• Figure 3: The contours for analytic continuation $z_1\to z_1+\varepsilon\omega_2$. The top/bottom figures show $\varepsilon=1$/$-1$. The left figures show the situation in the $u_1$ plane in the string limit while the right figures show the situation near the relativistic limit in the $q^{-2iu_1}$ plane where the left-hand branch points nearly touch at the origin. In the magnon theory $\theta_1=-\pi u_1/k$ and the top/bottom figures correspond to $\operatorname{Im}\theta_1$ decreasing/increasing so that in the relativistic limit the analytic continuation corresponds to $\theta_1\to\theta_1-i\varepsilon\pi$. The opposite is true in the mirror theory. The left figures show the sheets ${\cal R}_{n,0}$ that the contour crosses when starting on ${\cal R}_{0,0}$.
• Figure 4: The cuts $\text{x}^{(n)}_\pm$ in the $x$-plane for $n=1,2,3$ ($+$ in blue, $-$ in red) with $n$ increasing as the cuts move closer to the origin. The unit circle is shown in black. The three plots show the cuts in the (left) string limit with $g=1$, $k=50$; (middle) intermediate regime $k=20$, $g=3$; (right) relativistic limit $k=8$, $g=15$. In the relativistic limit $g\to\infty$ the left-hand branch points coalesce at $x=-1$ and the region of interest is the circular neighbourhood of $x=-1$.
• Figure 5: The cut structure of $\sigma(z_1,z_2)$ in the $q^{-2iu}$ plane on the sheets ${\cal R}_{0,0}$, ${\cal R}_{1,0}$ and ${\cal R}_{2,0}$. The red and blue cuts are identified. Note that the black cuts on the sheet ${\cal R}_{1,0}$ corresponding to the cuts $\text{x}^{(n)}_+$, $n=2,3,\dots$ going anti-clockwise from the cut $|x_1^-|=1$ (which is $\text{x}^{(1)}_+$), and $\text{x}^{(n)}_-$, $n=1,2,\ldots$ going clockwise from the cut $|x_1^+|=1$, of $\chi(x_1^+,x_2^\pm)$ become out of reach in the relativistic limit when all the inner branch points coalesce at the origin and the outer ones go to infinity. Also shown is the path for the analytic continuation $z_1\to z_1+\omega_2$.