Strings in AdS_4 x CP^3: finite size spectrum vs. Bethe Ansatz

TL;DR

This work studies curvature corrections to the light-cone string spectrum of type IIA strings on AdS$_4$×CP$^3$ in the near-BMN limit, computing $1/R^2$ finite-size effects and matching them with the all-loop Gromov–Vieira BA. It extends the one- and two-particle analyses to the full degenerate two-oscillator light bosonic sector, diagonalizes the resulting mixing matrix, and provides exact results in $\lambda'$ up to $\mathcal{O}(\lambda'^4)$, which agree with BA predictions including auxiliary-root regularization. A key result is the unitarity-preserving regularization that yields $a_1=0$ in the interpolation function $h(\lambda)$, implying no $\mathcal{O}(\lambda^0)$ shift at strong coupling, and the exponentially small finite-size corrections. The BA and string-theory computations show agreement at the same order in $\lambda'$ and $1/J$, supporting the all-order equivalence between the near-plane-wave string spectrum and the BA framework in AdS$_4$/CFT$_3$, while suggesting avenues to extend the test to heavier modes and to the Y-/T-system formalism. Altogether, the paper strengthens evidence for integrability-based spectral methods in this duality and highlights regularization choices as crucial for unitarity and exact matching.

Abstract

We compute the first curvature corrections to the spectrum of light-cone gauge type IIA string theory that arise in the expansion of $AdS_4\times \mathbb{CP}^3$ about a plane-wave limit. The resulting spectrum is shown to match precisely, both in magnitude and degeneration that of the corresponding solutions of the all-loop Gromov--Vieira Bethe Ansatz. The one-loop dispersion relation correction is calculated for all the single oscillator states of the theory, with the level matching condition lifted. It is shown to have all logarithmic divergences cancelled and to leave only a finite exponentially suppressed contribution, as shown earlier for light bosons. We argue that there is no ambiguity in the choice of the regularization for the self-energy sum, since the regularization applied is the only one preserving unitarity. Interaction matrices in the full degenerate two-oscillator sector are calculated and the spectrum of all two light magnon oscillators is completely determined. The same finite-size corrections, at the order 1/J, where $J$ is the length of the chain, in the two-magnon sector are calculated from the all loop Bethe Ansatz. The corrections obtained by the two completely different methods coincide up to the fourth order in $λ' =λ/J^2$. We conjecture that the equivalence extends to all orders in $λ$ and to higher orders in 1/J.

Figures (1)

• Figure 1: Schematic diagram of the contributing sectors of string Hamiltonian