On circular strings in $(AdS_3 \times S^3)_{\varkappa}$

TL;DR

This paper constructs and analyzes circular string solutions in the $oldsymbol{ ext{AdS}_3 imes S^3}$ background with a $oldsymbol{oldsymbol{ uarkappa}}$ deformation, exploring their quantization and novel energy-oscillation relationships.

Contribution

It introduces new circular string solutions in the $oldsymbol{ uarkappa}$-deformed background and examines their quantization and limiting behaviors, revealing novel energy dependence features.

Findings

Identified various circular string solutions in the deformed background.

Performed Bohr-Sommerfeld quantization to characterize string states.

Discovered a new dependence of oscillation number on energy in the long string limit.

Abstract

The so called one-parameter (often called $\varkappa$) deformed $AdS$ string sigma models have attracted a lot of attention lately in the study of integrability in string theory. We construct various circular string solutions in the $(AdS_3 \times S^3)_{\varkappa}$ background and describe the characteristics of such solutions qualitatively. We study the Bohr-Sommerfeld like quantization for these string states to characterise the motion. Further we find a `long' string limit of such circular strings in the $\varkappa$-deformed $AdS_3$ and find a novel dependence of the oscillation number on the energy in the next to leading order expansion.