Finite-gap equations for strings on AdS_3 x S^3 x T^4 with mixed 3-form flux

TL;DR

The paper advances integrability methods for string theory on $AdS_3 \times S^3 \times T^4$ with mixed RR and NSNS flux by deriving finite-gap equations from a generalized $\mathbb{Z}_4$-symmetric Lax framework and connecting them to an all-loop Bethe ansatz built from the Hoare--Tseytlin S-matrix. It shows that in the thermodynamic limit the Bethe equations reproduce the finite-gap construction while also providing leading-order and one-loop corrections to the dressing phases, and it analyzes semiclassical spectra through BMN quantization and classical solutions such as circular strings and giant magnons, including finite-size effects. The results extend the standard AdS/CFT integrability toolkit to backgrounds with mixed flux, enabling non-perturbative insights and cross-checks via S-duality; they also pave the way for incorporating massless modes and applying to other AdS$_3$ backgrounds. Overall, the work synthesizes algebraic-curve, Bethe-ansatz, and semiclassical analyses to yield a coherent picture of the classical and semi-classical spectrum in mixed-flux AdS$_3$ string theory, with precise predictions for dressing phases and finite-size corrections.

Abstract

We study superstrings on AdS_3 x S^3 x T^4 supported by a combination of Ramond-Ramond and Neveu-Schwarz-Neveu-Schwarz three form fluxes, and construct a set of finite-gap equations that describe the classical string spectrum. Using the recently proposed all-loop S-matrix we write down the all-loop Bethe ansatz equations for the massive sector. In the thermodynamic limit the Bethe ansatz reproduces the finite-gap equations. As part of this derivation we propose expressions for the leading order dressing phases. These phases differ from the well-known Arutyunov-Frolov-Staudacher phase that appears in the pure Ramond-Ramond case. We also consider the one-loop quantization of the algebraic curve and determine the one-loop corrections to the dressing phases. Finally we consider some classical string solutions including finite size giant magnons and circular strings.

Figures (3)

• Figure 1: The quasi-momenta form two sets of four-sheeted Riemann surfaces, corresponding to the hatted and checked quasi-momenta. Classical solutions correspond to cuts connecting the different sheets (of the same type, namely hatted or checked). The physical excitations correspond to poles connecting the sheets which contain the middle node root. Totally there are eight bosonic and fermionic physical excitations with different polarizations which are depicted in the figure by blue and red wiggly lines respectively. The lines contain different Dynkin nodes which correspond to the excited Bethe roots. The two Riemann surfaces are related by the inversion $x\to 1/x$ symmetry. The $\pm 1$ to the left of the Dynkin diagrams correspond to the grading of the superalgebra, these are the $su(2)$ (left digram) and $sl(2)$ (right diagram) gradings. The dashed lines separate the AdS and sphere's sheets.
• Figure 2: Integration contours in the $n$ and $x$ planes. The blue wavy lines indicate the two square root branch cuts with brach points $n=n_{\pm}$ and $x = \mp i$. The solid red line is the integration contour, which in the $n$ plane is taken along a circle that has been deformed to avoid the branch cuts. This picks up the poles of the $\cot$ function, indicated in the figure by purple crosses along the real axis. For large $N$ the contour in the $x$ plane approaches the branch cuts from the outside.
• Figure 3: For large $N$ we can use $\cot(\pi n) \approx \mp i$ to rewrite the integrals so that the contour splits into two parts, one in the upper and one in the lower half plane. Compared to the contours depicted in figure \ref{['fig:int-contours']} the direction of integration is reversed in the upper (lower) half of the $n$ ($x$) plane. Following Gromov:2007cd we denote this integral by $\int_{-1/s}^{+s} = \frac{1}{2} \int_{\hat{\mathcal{D}}_+} + \frac{1}{2} \int_{\hat{\mathcal{D}}_-}$, where $\hat{\mathcal{D}}_{\pm}$ indicates the two halves of the contour shown in the figure.