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We use the Atiyah-Singer index theorem to derive the general form of theone-loop corrections to observables in asymptotically anti-de Sitter (AdS$_4$)supersymmetric backgrounds of abelian gauged supergravity. Using the method ofsupergravity localization combined with the factorization of the supergravityaction on fixed points (NUTs) and fixed two-manifolds (Bolts) we show that ananalogous factorization takes place for the one-loop determinants ofsupergravity fields. This allows us to propose a general fixed-point formulafor the logarithmic corrections to a large class of supersymmetric partitionfunctions in the large $N$ expansion of a given 3d dual theory. The correctionsare uniquely fixed by some simple topological data pertaining to a particularbackground in the form of its regularized Euler characteristic $\chi$, togetherwith a single dynamical coefficient that counts the underlying degrees offreedom of the theory.

Factorization of log-corrections in AdS$_4$/CFT$_3$ from supergravity localization