Sulla generalizzazione di un problema di estremizzazione geometrico proveniente dalla corrispondenza AdS/CFT

Two different extremal problems in Conformal Field Theories, namely $a$-maximisation and $c$-extremisation, have been interpreted via the AdS/CFT correspondence as procedures dual to the volume minimisation problem for Sasaki-Einstein manifolds and to an analogous problem for the so-called GK geometries, respectively. Taking inspiration from these results, we study supersymmetric $AdS_3 \times M_7$, $AdS_5 \times M_5$ and $AdS_5 \times M_6$ solutions of both type IIB and $D=11$ supergravity in an attempt to generalise such extremal problems. All these geometries are characterised by the existence of a canonical Killing vector field that generates a symmetry of the whole solution - metric and fluxes - and is interpreted as the geometric dual of the R-symmetry. We obtain the actions of the effective theories reduced to the internal manifolds, imposing the integrability conditions of the effective Killing spinor equations. We recover Sasaki-Einstein and GK as particular cases of the first two classes of solutions considered since in both cases the effective action is the correct functional to be extremised. However, this is not always the case when all fluxes are turned on, as we shall prove in the class of generic $AdS_3 \times M_7$ solutions of type IIB, preserving $\mathcal{N} = (0,2)$ supersymmetry. When the internal manifold is odd-dimensional, we show that the off-shell gravitational central charge, which is the extremal functional of the dual field theory, can always be expressed in a purely geometric fashion in terms of spinor bilinears defining a G-structure. In particular, this expression only depends on the canonical Killing vector and on the fundamental form associated to the metric on the transverse space, which is either K\"ahler or balanced. For $AdS_5 \times M_6$ solutions of $D=11$ supergravity, even though the off-shell gravitational central charge is known to be expressed naturally in terms of an invariant of generalised geometry, we were not able to find an expression that only depends on spinor bilinears. We check our results with some known examples.