Finite-size corrections in the SU(2) x SU(2) sector of type IIA string theory on AdS_4 x CP^3

TL;DR

This work analyzes finite-size corrections in the SU(2)×SU(2) sector of type IIA string theory on AdS4×CP3, corresponding to ABJM theory. It employs two independent approaches—the curvature-corrected Penrose limit (1/J expansion) and a low-energy sigma-model expansion (λ' = λ/J^2)—to compute Δ−J corrections for R×S^2×S^2 states with large angular momentum J, finding remarkable agreement between methods and with the all-loop Bethe ansatz. Up to λ' and λ'^2 the results mirror the SU(2) sector of AdS5×S^5, while at λ'^3 interactions between the two S^2s emerge, leading to nontrivial finite-size effects; the results are also in accord with the Gromov–Vieira all-loop Bethe equations for small λ', including higher-order corrections. Together, these findings provide a stringent cross-check of integrability in the AdS4/CFT3 correspondence and of the proposed all-loop Bethe ansatz for ABJM theory.

Abstract

We consider finite-size corrections in the SU(2) x SU(2) sector of type IIA string theory on AdS_4 x CP^3, which is the string dual of the recently constructed N=6 superconformal Chern-Simons theory of Aharony, Bergman, Jafferis and Maldacena (ABJM theory). The string states we consider are in the R x S^2 x S^2 subspace of AdS_4 x CP^3 with an angular momentum J on CP^3 being large. We compute the finite-size corrections using two different methods, one is to consider curvature corrections to the Penrose limit giving an expansion in 1/J, the other by considering a low energy expansion in lambda'=lambda/J^2 of the string theory sigma-model, lambda being the 't Hooft coupling of the dual ABJM theory. For both methods there are interesting issues to deal with. In the near-pp-wave method there is a 1/\sqrt{J} interaction term for which we use zeta-function regularization in order to compute the 1/J correction to the energy. For the low energy sigma-model expansion we have to take into account a non-trivial coupling to a non-dynamical transverse direction. We find agreement between the two methods. At order lambda' and lambda'^2, for small lambda', our results are analogous to the ones for the SU(2) sector in type IIB string theory on AdS_5 x S^5. Instead at order lambda'^3 there are interactions between the two two-spheres. We compare our results with the recently proposed all-loop Bethe ansatz of Gromov and Vieira and find agreement.