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