Comments on ABJM free energy on S3 at large N and perturbative expansions in M-theory and string theory

We compare large N expansion of the localization result for the free energy F in the 3d N=6 superconformal U(N)k×U(N)−k Chern-Simons-matter theory to its AdS/CFT counterpart, i.e. to the perturbative expansion of M-theory partition function on AdS×4S7/Zk and to the weak string coupling expansion of type IIA effective action on AdS×4CP3. We show that the general form of the perturbative expansions of F on the two sides of the AdS/CFT duality is indeed the same. Moreover, the transcendentality properties of the coefficients in the large N, large k expansion of F match those in the corresponding M-theory or string theory expansions. To shed light on the structure of the 1-loop M-theory partition function on AdS×4S7/Zk we use the expression for the 1-loop 4-graviton scattering amplitude in the 11d supergravity. We also use the known information about the transcendental coefficients of the leading curvature invariants in the low-energy effective action of type II string theory. Matching of the remaining rational factors in the coefficients requires a precise information about currently unknown RR field strength terms in the corresponding superinvariants.