Stringing Spins and Spinning Strings

TL;DR

Beisert, Minahan, Staudacher, and Zarembo study planar one-loop anomalous dimensions of long scalar operators in N=4 SYM by mapping to an integrable Heisenberg spin chain. They identify two Bethe-root configurations in the SO(6) [J,L-2J,J] sector: a two-contour ground state with a smaller anomalous dimension and an all-imaginary-root state that reproduces the Frolov-Tseytlin semiclassical string predictions, including the fluctuation spectrum. The imaginary-root solution is shown to be the gauge dual of the semiclassical spinning string, while the double-contour state does not match, highlighting multiple competing semiclassical interpretations. An SO(6) singlet analogue is also found, yielding γ = 1/(4L) at α = 1, consistent with a pulsating circular string on S^5. Together, these results demonstrate a rich gauge/string duality structure in a high-density impurity regime and showcase the power of the Bethe ansatz in computing anomalous dimensions beyond BMN-like limits.

Abstract

We apply recently developed integrable spin chain and dilatation operator techniques in order to compute the planar one-loop anomalous dimensions for certain operators containing a large number of scalar fields in N =4 Super Yang-Mills. The first set of operators, belonging to the SO(6) representations [J,L-2J,J], interpolate smoothly between the BMN case of two impurities (J=2) and the extreme case where the number of impurities equals half the total number of fields (J=L/2). The result for this particular [J,0,J] operator is smaller than the anomalous dimension derived by Frolov and Tseytlin [hep-th/0304255] for a semiclassical string configuration which is the dual of a gauge invariant operator in the same representation. We then identify a second set of operators which also belong to [J,L-2J,J] representations, but which do not have a BMN limit. In this case the anomalous dimension of the [J,0,J] operator does match the Frolov-Tseytlin prediction. We also show that the fluctuation spectra for this [J,0,J] operator is consistent with the string prediction.

Figures (4)

• Figure 1: Bethe roots. For large $L$ the roots condense into two cuts
• Figure 2: Cuts flip for $\alpha<0$. Both cuts can be mapped to one $\mathcal{C}^\pm$ via the map $u^2=A+Bx$.
• Figure 3: States of lowest energy in $[J,L-2J,J]$, $J$ even. The plot shows how the energy per spin flip increases from the dilute gas, $\alpha=0$, to the maximum filling $\alpha=1/2$. Discrete values for $L\leq 19$ are obtained as eigenvalues of the dilatation generator and the curve for $L\to\infty$ represents the solution to the Bethe equations. The states with $J=2$ were found in Beisert:2002tn.
• Figure 4: Distribution of roots for the odd ground state.