Fibred GK geometry and supersymmetric $AdS$ solutions

TL;DR

This paper explores a broad class of supersymmetric AdS solutions with fibre geometries, providing a unified method to compute physical quantities like central charge and entropy without explicit metric details.

Contribution

It introduces a novel framework linking flux quantization and supersymmetric data to the 'master volume' of fibre geometries, enabling calculations of holographic quantities.

Findings

Derived formulas for flux quantization conditions.

Unified approach to compute central charge and entropy.

Confirmed results with known supergravity solutions and obtained new insights.

Abstract

We continue our study of a general class of $\mathcal{N}=2$ supersymmetric $AdS_3\times Y_7$ and $AdS_2\times Y_9$ solutions of type IIB and $D=11$ supergravity, respectively. The geometry of the internal spaces is part of a general family of "GK geometries", $Y_{2n+1}$, $n\ge 3$, and here we study examples in which $Y_{2n+1}$ fibres over a K\"ahler base manifold $B_{2k}$, with toric fibres. We show that the flux quantization conditions, and an action function that determines the supersymmetric $R$-symmetry Killing vector of a geometry, may all be written in terms of the "master volume" of the fibre, together with certain global data associated with the K\"ahler base. In particular, this allows one to compute the central charge and entropy of the holographically dual $(0,2)$ SCFT and dual superconformal quantum mechanics, respectively, without knowing the explicit form of the $Y_7$ or $Y_9$ geometry. We illustrate with a number of examples, finding agreement with explicit supergravity solutions in cases where these are known, and we also obtain new results. In addition we present, en passant, new formulae for calculating the volumes of Sasaki-Einstein manifolds.