Finite-size corrections in the SU(2) x SU(2) sector of type IIA string theory on AdS(4) x CP**3

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/J ^2 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.