Semiclassical circular strings in AdS_5 and "long" gauge field strength operators

TL;DR

This work investigates semiclassical strings in $AdS_5$ as duals to long gauge-theory operators built from self-dual field-strength components, where 1-loop anomalous dimensions map to an integrable $XXX_1$ spin chain. Focusing on circular strings with equal spins $S_1=S_2$ and the corresponding self-dual sector, the authors show that both the classical energy $E_0$ and the 1-loop correction $E_1$ scale linearly with the spin $S$ within a stability region, providing evidence for the duality with the ferromagnetic highest-spin state of the $XXX_1$ chain. They compute explicit results for the rigid rotating string and obtain $E_0 o 2.5 S + 0.65\sqrt{\lambda}$ and $E_1 \to -0.9\frac{S}{\sqrt{\lambda}}-2.38$ (for $m=1$), yielding a total energy linear in $S$ with coefficient $f(\lambda) \approx 2.5$ at large $\lambda$. Extending to rotating and pulsating configurations with $S_1=S_2$, the paper shows $E$ remains linear in both $S$ and the oscillation number $N$ in the regime ${\cal S},{\cal N}=O(1)$, with several asymptotic limits (e.g., $E\sim 2\lambda^{1/4}\sqrt{mL}$ for small $S,N$ and $E\sim 2L+\cdots$ for large $N$), and identifies the zero-spin pulsating case with the antiferromagnetic vacuum of the same spin chain. These results reinforce the AdS/CFT picture beyond fast-string regimes and illuminate how different string solutions correspond to distinct sectors of the spin-chain spectrum.

Abstract

We consider circular strings rotating with equal spins S_1=S_2=S in two orthogonal planes in AdS_5 and suggest that they may be dual to "long" gauge theory operators built out of self-dual components of gauge field strength. As was found in hep-th/0404187, the one-loop anomalous dimensions of the such gauge-theory operators are described by an anti-ferromagnetic XXX_1 spin chain and scale linearly with length L>>1. We find that in the case of rigid rotating string both the classical energy E_0 and the 1-loop string correction E_1 depend linearly on the spin S (within the stability region of the solution). This supports the relation between the rigid rotating string and the gauge-theory operator corresponding to the maximal-spin (ferromagnetic) state of the XXX_1 spin chain. The energy of more general rotating and pulsating strings also happens to scale linearly with both the spin and the oscillation number. Such solutions should be dual to other lower-spin states of the spin chain, with the anti-ferromagnetic ground state presumably corresponding to the string pulsating in two planes with no rotation.

Figures (3)

• Figure 1: Plot of the classical energy ${\cal E}({\cal S})$ for $m=1$. The dashed line is $2.5\mathcal{S}+ 0.65$.
• Figure 2: One-loop correction to energy $E_{1}$ as a function of $\mathcal{S}$
• Figure 3: $\mathcal{N} ({\cal E})$ for $\mathcal{S}=0.4$