Geometries with Killing Spinors and Supersymmetric AdS Solutions

TL;DR

This work introduces a broad class of geometries in $2n+2$ dimensions that admit a Killing spinor and generalize Sasaki–Einstein-type structures, linking them to supersymmetric AdS solutions. The authors develop an $SU(n{+}1)$ structure framework with metric, scalar, and closed 3-form, derive the corresponding equations of motion from a Lagrangian, and show that compact solutions require $f=0$ and $\phi=0$ (Ricci-flat holonomy), while non-compact cones lead to rich $2n{+}1$-dimensional base geometries with closed 2-forms. They present three concrete avenues to generate explicit examples: (i) a fibration construction over positively curved KE$^+_{2n-2}$ spaces, yielding infinite families of smooth compact $2n{+}1$ geometries; (ii) a multi-factor ansatz for $2n+2$ that recovers AdS$_3$ solutions in IIB (n=3) and AdS$_2$ solutions in D=11 (n=4), including interpolations between AdS regions; (iii) an LLM-inspired generalization to arbitrary $n$, producing a master equation $\Delta D + x^{(n-4)/(n-3)}\partial_x^2 e^D=0$ (reducing to the linear LLM equation at $n=3$ and to a continuous Toda-type equation at $n=4$). The results offer a unified geometric framework for constructing new supersymmetric AdS solutions and point to rich structures and potential dual field theories, particularly in the $n=3,4$ cases.

Abstract

The seven and nine dimensional geometries associated with certain classes of supersymmetric $AdS_3$ and $AdS_2$ solutions of type IIB and D=11 supergravity, respectively, have many similarities with Sasaki-Einstein geometry. We further elucidate their properties and also generalise them to higher odd dimensions by introducing a new class of complex geometries in $2n+2$ dimensions, specified by a Riemannian metric, a scalar field and a closed three-form, which admit a particular kind of Killing spinor. In particular, for $n\ge 3$, we show that when the geometry in $2n+2$ dimensions is a cone we obtain a class of geometries in $2n+1$ dimensions, specified by a Riemannian metric, a scalar field and a closed two-form, which includes the seven and nine-dimensional geometries mentioned above when $n=3,4$, respectively. We also consider various ansatz for the geometries and construct infinite classes of explicit examples for all $n$.