Factorization of log-corrections in AdS4/CFT3 from supergravity localization

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