Exploring the Quantum Spectral Curve for AdS${}_3$/CFT${}_2$

TL;DR

This work demonstrates that the recently proposed Quantum Spectral Curve for AdS$_3$/CFT$_2$ with pure RR flux yields concrete, high-precision predictions for non-protected string excitations at finite tension. By confronting non-quadratic branch cuts introduced by massless modes, the authors develop new analytic tools (notably the quadratic-cut $\mathbb{P}$-functions) and tailored numerical algorithms that extend the QSC framework beyond the AdS$_5$ paradigm. They obtain both a systematic weak-coupling expansion up to $g^8$ and high-precision numerical spectra for twists $S=2,4,6$, providing nontrivial data to probe the yet-mysterious dual CFT and to test integrability-based conjectures. The results illuminate the role of massless modes in 2D CFT holography and open avenues for bootstrability-like approaches and exact computations of correlation data in this class of AdS$_3$/CFT$_2$ dualities.

Abstract

Despite the rich and fruitful history of the integrability approach to string theory on the $AdS_3\times S^3\times T^4$ background, it has not been possible to extract many concrete predictions from integrability, except in a strict asymptotic regime of large quantum numbers, due to the severity of wrapping effects. The situation changed radically with two independent and identical proposals for the Quantum Spectral Curve (QSC) for this system in a background of pure Ramond-Ramond flux. This formulation is expected to capture all wrapping effects exactly and describe the full planar spectrum. Massless modes conjecturally manifest themselves in a new property of this QSC: the non-quadratic nature of the branch-cut singularities of the QSC Q-functions. This feature implies new technical challenges in solving the QSC equations as compared to the well-studied case of N=4 SYM. In this paper we resolve these difficulties and obtain the first ever predictions for generic unprotected string excitations. We explain how to extract a systematic expansion around the analogue of the weak 't Hooft coupling limit in N=4 SYM and also obtain high-precision numerical results. This concrete data and others obtainable from the QSC could help to identify the so-far mysterious dual CFT.

Figures (4)

• Figure 1: Two non-identical analytic continuations around the branch point at $u=-2g$ we denote by $\gamma$ (clock-wise) and $\bar{\gamma}$ (anti-clock-wise).
• Figure 2: In the AdS${}_5$ case, the cut of the ${\bf P}$-functions is quadratic, which results in the large convergence domain (blue on the left figure) with inner radius going to zero as $g$ at weak coupling. In the current case the convergence domain stops at the remaining non-rationalized branch points at $\pm 1$ so that the inner convergence radius of the Laurent series is $1$.
• Figure 3: Parasitic cuts in the function ${\mathbb P}_a$ constrain the convergence domain (in green) to be the annulus with the inner radius scaling as $g^{1/3}$ and the outer radius scaling as $g^{-1/3}$. Importantly, the Laurent series for the function ${\mathbb P}_a$ converges fast in the vicinity of the unit circle, allowing to perform the gluing procedure for the functions living inside and outside the unit circle, such as ${\bf Q}_k$ and ${\bf Q}^\gamma_{\dot k}$.
• Figure 4: Numerical fit of $\gamma$ for $S=2$, $S=4$ and $S=6$. The bottom curve corresponds to $S=2$, the middle to $S=4$ and the top curve to $S=6$.