More on the tensionless limit of pure-Ramond-Ramond AdS3/CFT2

TL;DR

The paper investigates the tensionless (h -> 0) limit of pure-RR AdS3xS3xT4 string theory using mirror TBA and contour-deformation methods to describe the excited-state spectrum. It derives and simplifies the excited-state equations, decoupling massless and auxiliary sectors at leading order, and demonstrates that the leading anomalous dimensions arise from massless modes with energy scaling as O(h). Numerical solutions for states with two and four excitations show energies approaching a free, weakly interacting massless-magnon behavior, with finite-size corrections scaling as 1/L. The study also analyzes the role of the N0 parameter (types of massless excitations) and provides a detailed numerical algorithm, including regularisation of kernels and exact Bethe equations, to obtain high-precision spectra. The results suggest the tensionless dynamics are not a simple nearest-neighbour spin chain nor a symmetric orbifold CFT, but a distinct long-range, massless-magnon regime with rich integrable structure and potential links to Y-system/Cartan-like formulations.

Abstract

In a recent letter we presented the equations which describe tensionless limit of the excited-state spectrum for strings on $AdS_3\times S^3\times T^4$ supported by Ramond-Ramond flux, and their numerical solution. In this paper, we give a detailed account of the derivation of these equations from the mirror TBA equations proposed by Frolov and Sfondrini, discussing the contour-deformation trick which we used to obtain excited-state equations and the tensionless limit. We also comment at length on the algorithm for the numerical solution of the equations in the tensionless limit, and present a number of explicit numerical results, as well as comment on their interpretation.

Figures (11)

• Figure 1: Convergence of the exact Bethe root $\gamma_1^{[n]}$ at the $n$-th iteration for world-sheet volume $L=32$ and mode number $\nu=1$. Each dot represents an iteration. On the left the plot with the values of $\gamma_1^{[n]}$, while on the right the deviations with respect to the final value. The plot on the right has y-axis in log scale to emphasise the exponential convergence. The starting value is the asymptotic Bethe root, as explained in the previous subsection.
• Figure 2: Y functions for a state with $L=256$ and mode number $\nu=1$. On the left $Y_0(\gamma)$ and on the right $Y(\gamma)$. The red crosses indicate the positions of the Bethe roots, where both the Y functions are equal to zero and both change sign. $Y_0(\gamma)$ takes very small positive values in the interval $[-\gamma_1,\gamma_1]$, since it rapidly converges to zero as $\gamma\to 0$. For $|\gamma|\gg\gamma_1$ both Y functions quickly converge to their asymptotic values, see table \ref{['table:AsymYfunc']}. Even though the plots show only the region $|\gamma|<25$, the cutoff has been set to $\Lambda=40$.
• Figure 3: Anomalous dimensions $H_{(1)}$ at order $\order{h}$ in the string tension. The plots are for different values of L, namely $L=4,16,32,256$. In each plot are represented the exact energy from \ref{['weak exact energy']}, the asymptotic energy from the Bethe-Yang equations, and the energy for a free model, see \ref{['eq:freeapprox']}.
• Figure 4: Deviation of the asymptotic energy $H_{(1)}^{BY}$ from the exact energy $H_{(1)}$ for different values of the world-sheet length $L$. The deviation goes like $1/L$.
• Figure 5: Y functions for a state with $L=16$ and mode numbers $\nu_1=1$ and $\nu_3=4$. On the left $Y_0(\gamma)$ and on the right $Y(\gamma)$. The crosses indicate the positions of the 4 Bethe roots, where both the Y functions are equal to zero and both change sign. The red ones are associated to the modes $(-\nu_1,\nu_1)$ and the blue ones to the modes $(-\nu_3,\nu_3)$. In $Y_0$ the changes of sign due to the bigger Bethe roots (marked by the red crosses) are distinguishable, while the other two (marked by the blue crosses) are not, since again $Y_0(\gamma)$ rapidly converges to zero as $\gamma\to 0$. Again, the large $\gamma$ behaviour is the expected one and they quickly converge to their asymptotic values as stated in Table \ref{['table:AsymYfunc']}. The plots shows the region $\left| \gamma\right|<18$, but again the cutoff has been set to $\Lambda=40$.