Finite-size corrections in the [formula omitted] sector of type IIA string theory on [formula omitted

Abstract

We consider finite-size corrections in the SU ( 2 ) × SU ( 2 ) sector of type IIA string theory on AdS 4 × C P 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 × S 2 × S 2 subspace of AdS 4 × C P 3 with an angular momentum J on C P 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 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 AdS 5 × S 5 . 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.