[Paper Review] Light-Cone Gauge in Non-Relativistic AdS$_5 imes$S$^5$ String Theory

Andrea Fontanella, Juan Miguel Nieto García|arXiv (Cornell University)|Jan 29, 2021

Black Holes and Theoretical Physics28 references4 citations

TL;DR

This paper derives the non-relativistic limit of string theory in AdS₅×S⁵ using Cartesian coordinates and uniform light-cone gauge, showing that fluctuations around a twisted BMN-like string become free fields at large string tension and large AdS radius. The key result is the emergence of a free-field action in the large-R and large-T limit, confirming consistency and integrability in the non-relativistic regime.

ABSTRACT

We discuss the non-relativistic limit of string theory in AdS$_5 imes$S$^5$ for different choices of embedding coordinates. We show that, if we consider Cartesian coordinates, the action of fluctuations around the twisted BMN-like string found in arXiv:2109.13240 in uniform light-cone gauge becomes the one of free fields at large string tension and large AdS$_5$ radius.

Motivation & Objective

• To investigate the non-relativistic limit of string theory in AdS₅×S⁵ using Cartesian coordinates.
• To analyze the behavior of string fluctuations in the uniform light-cone gauge under large string tension and large AdS radius.
• To confirm that the effective action for small fluctuations becomes that of free fields in the large-R and large-T limit.
• To establish consistency of the non-relativistic string theory by verifying the role of winding modes via Lagrange multipliers.

Proposed method

• The authors perform a coordinate transformation to Cartesian global coordinates in AdS₅×S⁵, enabling a systematic expansion in the large string tension parameter T and large AdS radius R.
• They apply the uniform light-cone gauge to the non-relativistic string action derived via the String Newton-Cartan formalism.
• The action is expanded in powers of 1/ω, where ω is the large parameter analogous to the speed of light, with the critical B-field introduced to cancel divergences.
• The resulting perturbative expansion yields cubic and quartic Lagrangians that explicitly preserve SO(3)×SO(4) symmetry.
• The limit R→∞ and T→∞ is taken in the action, showing that higher-order terms vanish and the dynamics reduce to free fields.
• The analysis confirms that the longitudinal metric and boost-invariant tensor are consistent with String Newton-Cartan geometry, and the Lagrange multipliers enforce the winding condition.

Experimental results

Research questions

• RQ1 Does the non-relativistic limit of AdS₅×S⁵ string theory in Cartesian coordinates yield a consistent free-field action for small fluctuations in the large-T and large-R regime?
• RQ2 How does the uniform light-cone gauge simplify the non-relativistic string action in this limit?
• RQ3 What is the role of the critical B-field and Lagrange multipliers in maintaining consistency under the non-relativistic limit?
• RQ4 Are the cubic and quartic interaction terms in the effective action suppressed in the large-T and large-R limit, leading to free dynamics?
• RQ5 Does the SO(3)×SO(4) symmetry remain manifest in the non-relativistic limit, indicating preserved rotational invariance?

Key findings

• In the large string tension and large AdS radius limit, the action for fluctuations around the twisted BMN-like string reduces to a free-field theory.
• The cubic and quartic Lagrangians derived in the expansion are explicitly symmetric under SO(3)×SO(4), indicating preserved rotational invariance.
• The longitudinal metric τμν and boost-invariant tensor Hμν take a simple form in Cartesian coordinates, consistent with String Newton-Cartan geometry.
• The critical B-field compensates the divergent metric term, allowing a well-defined non-relativistic limit without introducing physical degrees of freedom.
• The Lagrange multiplier equations enforce the winding condition, confirming that a non-trivial spectrum requires string winding in the longitudinal direction.
• The perturbative expansion is well-defined at all orders, with no divergences, and the leading-order dynamics are free, supporting the consistency of the non-relativistic string theory.