A Spin Chain from String Theory

TL;DR

The paper studies the semiclassical spectrum of strings on $AdS_{3}\times S^{1}$ in the limit of large spin $S$, showing that these $K$-gap string states correspond to operators in the $sl(2)$ sector of ${\cal N}=4$ SYM described by a classical $SL(2,\mathbb{R})$ spin chain of length $K$, up to a universal cusp dimension prefactor $\Gamma(\lambda)$. Using finite-gap methods, the authors derive a degenerate spectral curve in the large-$S$ limit, reducing the string dynamics to a finite-dimensional spin-chain problem with $K$ sites; the energies satisfy $\Delta-S = (\sqrt{\lambda}/2\pi)[K\log S + \log(\tilde{q}_{K}/\sqrt{-\tilde{q}_{2}})]+\cdots$, and after quantization and suitable identifications this matches the gauge theory result $\gamma_{\text{one-loop}} = (\lambda/4\pi^{2})(K\log S + H_{K}[...]+O(1/S))$. This provides a precise spike-to-spin correspondence and demonstrates a mechanism by which a continuous string reduces to a discrete integrable system in the large-$S$ limit, while preserving the gauge/string duality at the quantum level through a universal coupling-dependent factor. The work thus reveals a universal, planar-field-theory-linked structure in the large-spin sector and suggests potential applications to broader contexts such as large-$N$ QCD.

Abstract

We study the semiclassical spectrum of bosonic string theory on AdS_3 x S^1 in the limit of large AdS angular momentum. At leading semiclassical order, this is a subsector of the IIB superstring on AdS_5 x S^5. The theory includes strings with K spikes which approach the boundary in this limit. We show that, for all K, the spectrum of these strings exactly matches that of the corresponding operators in the dual gauge theory up to a single universal prefactor which can be identified with the cusp anomalous dimension. We propose a precise map between the dynamics of the spikes and the classical SL(2,R) spin chain which arises in the large-spin limit of N=4 Super Yang-Mills theory.

Figures (8)

• Figure 1: The cut $x$-plane corresponding to the curve $\Gamma_{K}$
• Figure 2: The cut $x$-plane corresponding to the curve $\Sigma$
• Figure 3: The cycles on $\Sigma$. The index $I$ runs from $1$ to $K/2$.
• Figure 4: The degeneration $\Sigma \rightarrow \tilde{\Sigma}_{1}\cup \tilde{\Sigma}_{2}$. The four singular points $\pm 1^{\pm}$ are marked with crosses on the curve
• Figure 5: The cut $\tilde{x}$-plane corresponding to the curve $\tilde{\Sigma}_{1}$