One-loop supergravity on AdS₄×S⁷/ℤ_k and comparison with ABJM theory

The large-N limit of ABJM theory is holographically dual to M-theory on AdS₄×S⁷/ℤ_k. The 3-sphere partition function has been obtained via localization, and its leading behavior F_ABJM⁽⁰⁾∼k^1/2N^3/2 is exactly reproduced in the dual theory by tree-level supergravity. We extend this comparison to the sub-leading O(N⁰) order by computing the one-loop supergravity free energy as a function of k and comparing it with the ABJM result. Curiously, we find that the expressions do not match, with F_SUGRA⁽¹⁾∼k⁶, while F_ABJM⁽¹⁾∼k². This suggests that the low-energy approximation Z_M-theory=Z_SUGRA breaks down at one-loop order.