A new coordinate system for q-deformed AdS_5 x S^5 and classical string solutions

We study a GKP-like classical string solution on a q-deformed AdS_5 x S^5 background and argue the spacetime structure by using it as a probe. The solution cannot stretch beyond the singularity surface and this result may suggest that the holographic relation is realized inside the singularity surface. This observation leads us to introduce a new coordinate system which describes the spacetime only inside the singularity surface. With the new coordinate system, we study minimal surfaces and derive a deformed Neumann-Rosochatius system with a rigid string ansatz.

💡 Research Summary

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The paper investigates classical string dynamics on the q‑deformed AdS₅×S⁵ background introduced by Arutyunov, Borsato and Frolov (ABF). The metric and NS‑NS two‑form are known, while the dilaton and RR fields remain undetermined. A key feature of the deformed AdS₅ part is the presence of a singularity surface at ρₛ = arcsinh(1/C), where C ≥ 0 is the deformation parameter. Beyond this surface the sign of the time component of the metric flips, rendering the region unphysical for string propagation.

The authors first consider a GKP‑type folded rotating string with the ansatz t = κτ, ψ₁ = ωτ, ρ = ρ(σ) and all other coordinates set to zero. The Wess‑Zumino term vanishes for this ansatz, so the dynamics are governed solely by the deformed metric. Solving the equations of motion together with the Virasoro constraints yields an effective first‑order equation for ρ′² (eq. 2.10). The solution oscillates between two turning points ρ₍₋₎⁰ and ρ₍₊₎⁰, which are functions of κ, ω and C (eq. 2.12). Reality and positivity of these turning points impose the inequality ω ≥ (√(1+C²)+C) κ. Importantly, both turning points lie strictly below the singularity surface, i.e. ρ₍₊₎⁰ < ρₛ. Consequently, the folded string cannot extend beyond the singularity. This observation suggests that the holographic correspondence, if it exists, must be confined to the interior of the singularity surface.

Motivated by this, the authors introduce a new coordinate system that covers only the region 0 ≤ ρ < ρₛ. In these coordinates the metric remains regular and the problematic region with flipped time signature is excised. Using the new coordinates they construct minimal surfaces. The simplest static solution describes a deformed AdS₂ subspace; in the limit C → 0 it reduces to the familiar global AdS₂ surface. Adding angular momentum yields a deformed version of the known rotating minimal surface, again smoothly reducing to the undeformed case when C vanishes.

The paper then turns to the S⁵ sector. By imposing a rigid‑string ansatz (constant angular velocities on the internal angles) the authors reduce the dynamics to a one‑dimensional integrable system of Neumann‑Rosochatius (NR) type. The deformation introduces C‑dependent trigonometric and hyperbolic potentials, preserving integrability while modifying the interaction structure. The resulting Lagrangian (section 4) can be brought to a canonical form with explicit C‑dependent coefficients.

To make the energy–spin relation explicit, two limiting regimes are examined. In the short‑string limit (ω ≫ κ) the turning point ρ₍₋₎⁰ ≈ κ/ω is small, allowing an expansion of the conserved charges. The resulting relation is E² ≈ 2√λ (1+C²)^{1/2} S + …, which reproduces the classic GKP result when C → 0, up to overall C‑dependent factors. In the long‑string limit the string stretches close to the singularity; after a change of variables the periodicity condition leads to an integral involving elliptic functions. The energy acquires a logarithmic term reminiscent of the undeformed long‑string behavior, but with C‑dependent modifications in the coefficient and argument of the log.

Appendices provide additional material: Appendix A discusses geodesics of massless and massive particles in both the original ABF and the new coordinates, confirming that the singularity surface acts as a causal boundary. Appendix B re‑derives the giant‑magnon solution in the new coordinates, showing that the dispersion relation remains unchanged. Appendix C derives a deformed NR system from the AdS₅ part, yielding hyperbolic potentials analogous to those obtained from the S⁵ sector.

Overall, the work clarifies the geometric role of the singularity surface in q‑deformed AdS₅×S⁵, demonstrates that classical string solutions are naturally confined within it, and provides a convenient coordinate framework for further investigations. The deformed NR system and the explicit energy–spin formulas constitute valuable tools for future studies of the quantum spectrum and potential holographic duals of the q‑deformed background.