Comment on Strings in AdS3 x S3 x S3 x S1 at One Loop

TL;DR

This work develops an algebraic‑curve treatment of semiclassical strings in $AdS_3 \times S_3 \times S_3 \times S^1$ to compute one‑loop energy corrections for giant magnons and long spinning strings. By comparing corrections across the giant magnon and spinning-string sectors, the authors derive a universal prediction for the one‑loop cusp term $f_1$ in $f(h)$ and fix the constant $c$ in $h(\lambda)$, while exposing strong regularisation dependence and pathologies in certain cutoffs as $\alpha \to 1$. They find a magnon‑based expression for $f_1$ that differs from the naive AdS$_5$/AdS$_4$ expectations, indicating a distinct dressing phase in this background. The results also connect to recent proposals for the $AdS_3 \times S_3 \times S_3 \times S^1$ S‑matrix and highlight the crucial role of massless modes and finite‑size effects in this interpolating theory.

Abstract

This paper studies semiclassical strings in AdS3 x S3 x S3 x S1 using the algebraic curve. Calculating one-loop corrections to the energy of the giant magnon fixes the constant term c in the expansion of the coupling h(λ). Comparing these to similar corrections for long spinning strings gives a prediction for the one-loop term f_1 in the expansion of the cusp anomalous dimension f(h), for all α(where α--> 1 is the AdS3 x S3 x T4 limit). For these semiclassical mode sums there is a similar choice of regularisation prescriptions to that encountered in AdS4 x CP3. However at α\neq 1/2 they lead to different values of f_1 and are therefore not related by a simple change of the coupling. The algebraic curve is also used to calculate various finite-size corrections for giant magnons, which are well-behaved as α--> 1, and can be compared to the recently published S-matrices.