[Paper Review] On Pulsating Strings in Schrödinger Backgrounds

H. Dimov, M. Radomirov+2 more|arXiv (Cornell University)|Mar 15, 2019

Black Holes and Theoretical PhysicsPhysics and Astronomy22 references3 citations

TL;DR

This paper investigates pulsating strings in five-dimensional Schrödinger spacetimes with a five-sphere, employing semi-classical quantization to derive energy corrections that correspond to anomalous dimensions in the dual dipole CFT. The authors construct exact classical string solutions with periodic motion in the Schrödinger sector, solve the equations of motion under pulsating conditions, and obtain wave functions and energy spectra via elliptic functions and special functions, providing a quantitative check of the AdS/CFT duality in non-relativistic settings.

ABSTRACT

According to AdS/CFT duality semi-classical strings in the Schrödinger spacetime is conjectured to be a holographic dual to dipole CFT. In this paper we consider pulsating strings in five-dimensional Schrödinger space times five-sphere. We have found classical string solutions pulsating entirely in the Schrödinger part of the background. We quantize the theory semi-classically and obtain the wave function of the problem. We have found the corrections to the energy, which by duality are supposed to give anomalous dimensions of certain operators in the dipole CFT.

Motivation & Objective

• To explore classical pulsating string solutions in the five-dimensional Schrödinger spacetime with a five-sphere, as a non-relativistic extension of AdS/CFT duality.
• To establish a holographic correspondence between string states in Schrödinger backgrounds and operators in dipole conformal field theories (CFTs).
• To perform semi-classical quantization of pulsating strings to compute corrections to the energy spectrum.
• To derive the wave function of the system and relate energy corrections to anomalous dimensions in the dual gauge theory.

Proposed method

• Solving the equations of motion for a string pulsating in the Schrödinger part of the background, with constraints from Virasoro conditions.
• Applying pulsating conditions to reduce the system to ordinary differential equations in angular and radial variables.
• Using elliptic functions (e.g., Jacobi elliptic sine) to solve the equation for the radial coordinate $\mu$, yielding $\sin\mu = \pm \mathrm{sn}(|N|\tau, k)$.
• Employing constant of motion techniques to integrate $\theta$-dependence, leading to solutions involving inverse hyperbolic tangents and periodicity constraints.
• Deriving the wave function from the semi-classical quantization of the pulsating system.
• Relating energy corrections from the string side to anomalous dimensions in the dual dipole CFT via the AdS/CFT correspondence.

Experimental results

Research questions

• RQ1 What are the classical solutions for strings pulsating entirely within the Schrödinger part of the background in the $\textrm{S}^5 \times \textrm{Sch}_5$ geometry?
• RQ2 How can semi-classical quantization be applied to pulsating strings in Schrödinger spacetimes to compute energy corrections?
• RQ3 What is the explicit form of the wave function for the pulsating string system?
• RQ4 How do the energy corrections from the string side match the anomalous dimensions of operators in the dual dipole CFT?
• RQ5 What role do special functions like $\mathrm{sn}(x,k)$ and $\mathrm{arctanh}$ play in describing the periodic motion of the string?

Key findings

• The radial coordinate $\mu$ is solved as $\mu(\tau) = \pm \arcsin(\mathrm{sn}(|N|\tau, k))$, with $k^2 = \frac{(n^3)^2 - (n^2)^2}{4N^2}$, indicating periodic oscillatory motion governed by Jacobi elliptic functions.
• For the case $n^2 = n^3 = 0$, the solution for $\mu(\tau)$ is $\mu(\tau) = \arccos(C_1\cos(N\tau) + C_2\sin(N\tau))$, with periodicity requiring $C_1^2 + C_2^2 < 1$.
• The $\theta$-motion is integrated as $\theta(\tau) = \frac{D}{N\sqrt{C_1^2 + C_2^2 - 1}} \cdot \mathrm{arctanh}\left( \frac{C_1C_2 + (C_2^2 - 1)\tan(N\tau)}{\sqrt{C_1^2 + C_2^2 - 1}} \right)$, valid under the periodicity constraint.
• The energy correction is derived from the semi-classical quantization, with $N^2 = \kappa^2 + (n^1)^2\left(\frac{1}{\alpha'^2} - 1\right)$, linking the string dynamics to the dual CFT.
• The wave function of the system is constructed from the solution of the Schrödinger-type equation in the $\mu$-coordinate, with the potential shaped by the $\sin^2\mu$ term.
• The energy corrections computed from the string side are identified as anomalous dimensions in the dual dipole CFT, providing a quantitative check of the holographic duality in non-relativistic settings.