Finite-size corrections in the SU(2)×SU(2) sector of type IIA string theory on AdS4×CP3

We consider finite-size corrections in the SU(2)×SU(2) sector of type IIA string theory on AdS4×CP3, 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×S2×S2 subspace of AdS4×CP3 with an angular momentum J on CP3 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 λ′=λ/J2 of the string theory sigma-model, λ 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/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 λ′ and λ′2, for small λ′, our results are analogous to the ones for the SU(2) sector in type IIB string theory on AdS5×S5. Instead at order λ′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.