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We consider finite-size corrections in the SU(2) x SU(2) sector of type IIAstring theory on AdS_4 x CP^3, which is the string dual of the recentlyconstructed N=6 superconformal Chern-Simons theory of Aharony, Bergman,Jafferis and Maldacena (ABJM theory). The string states we consider are in theR x S^2 x S^2 subspace of AdS_4 x CP^3 with an angular momentum J on CP^3 beinglarge. We compute the finite-size corrections using two different methods, oneis to consider curvature corrections to the Penrose limit giving an expansionin 1/J, the other by considering a low energy expansion in lambda'=lambda/J^2of the string theory sigma-model, lambda being the 't Hooft coupling of thedual 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 weuse zeta-function regularization in order to compute the 1/J correction to theenergy. For the low energy sigma-model expansion we have to take into account anon-trivial coupling to a non-dynamical transverse direction. We find agreementbetween the two methods. At order lambda' and lambda'^2, for small lambda', ourresults are analogous to the ones for the SU(2) sector in type IIB stringtheory on AdS_5 x S^5. Instead at order lambda'^3 there are interactionsbetween the two two-spheres. We compare our results with the recently proposedall-loop Bethe ansatz of Gromov and Vieira and find agreement.

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