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

Abstract

We use the Atiyah-Singer index theorem to derive the general form of the one-loop corrections to observables in asymptotically anti-de Sitter (AdS4) 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 χ, together with a single dynamical coefficient that counts the underlying degrees of freedom of the theory.

4d $$ \mathcal{N} $$ = 2 supergravity observables from Nekrasov-like partition functions

“…What is perhaps most surprising in such an identification is the conclusion that the off-shell localization locus in the supergravity localization formalism can be mapped to the on-shell space of Coulomb branch solutions via the simple φ I = χ I . Such an identification provides a natural extension of the supergravity localization program to the cases of black holes with rotation, for which the 1-loop contribution at each fixed point is already known as well [48].…”

“…formula (1.3). The existence of a smooth limit between nuts and bolts, emphasized in [47,48], allow us to also infer the contribution from a bolt, see eq. (4.37) and the discussion around it.…”

“…leave this as an open problem, which needs to be addressed after more evidence accumulates for the validity of part II of the conjecture. Provided a clear identification exists, the results of [47,48] about the one-loop correction give an explicit example of what the general Nekrasov partition function Z UV Nek should include, namely the loop corrections arising both from the massless four-dimensional effective action and from the full infinite set of massive KK modes. This is precisely in line with the discussion below eq.…”

“…Due to the discovery of the Coulomb branch solutions [36], again in the STU model, converting to the microcanonical ensemble no longer needed for a precise holographic match. It would therefore be interesting to explore fully the implications from the full holographic identification of (4.46) with (4.47) and in turn (4.48) with (4.49) beyond the leading order, extending the available subleading corrections discussed in [13,48,152,153,[157][158][159][160][161]. The identification of these grand-canonical partition functions would of course immediately imply a precise match of the respective microcanonical quantities, but is a slightly simpler task due to lack of additional integration over the chemical potentials.…”

Wrapped M5-branes and complex saddle points

“…One of the far-reaching properties of the formula for the on-shell action is that we don't need the analytic form of the metric, which is generally quite difficult to find, but rather only knowledge of the topology of Y 4 and of the circle action generated by ξ. This poses conceptual problems, such as suggesting the existence of an underlying fixed point theorem acting on the supergravity background, especially given that this localization persists also for corrections to the two-derivative model [10][11][12], and it also applies to complex metrics [13]. More concretely, it provides a way of computing the on-shell action of any smooth solution, assuming its existence.…”