One-loop supergravity on AdS4 x S7 / Zk and comparison with ABJM theory

The large-N limit of ABJM theory is holographically dual to M-theory on AdS(4) x S-7 / Z(k). The 3-sphere partition function has been obtained via localization, and its leading behavior F-ABJM((0)) similar to k(1/2) N-3/2 is exactly reproduced in the dual theory by tree- level supergravity. We extend this comparison to the sub-leading O (N-0) order by computing the one-loop supergravity free energy as a function of k and comparing it with the ABJM result. Curiously, we fi nd that the expressions do not match, with F-SUGRA((1)) similar to k(6), while F-ABJM((1))similar to k(2). This suggests that the low- energy approximation Z(M-theory) = Z(SUGRA) breaks down at one-loop order.