Pseudo-Killing Spinors, Pseudo-supersymmetric p-branes, Bubbling and Less-bubbling AdS Spaces

We consider Einstein gravity coupled to an n-form field strength in D dimensions. Such a theory cannot be supersymmetrized in general, we nevertheless propose a pseudo-Killing spinor equation and show that the AdS X Sphere vacua have the maximum number of pseudo-Killing spinors, and hence are fully pseudo-supersymmetric. We show that extremal p-branes and their intersecting configurations preserve fractions of the pseudo-supersymmetry. We study the integrability condition for general (D,n) and obtain the additional constraints that are required so that the existence of the pseudo-Killing spinors implies the Einstein equations of motion. We obtain new pseudo-supersymmetric bubbling AdS_5 X S^5 spaces that are supported by a non-self-dual 5-form. This demonstrates that non-supersymmegtric conformal field theories may also have bubbling states of arbitrary droplets of free fermions in the phase space. We also obtain an example of less-bubbling AdS geometry in D=8, whose bubbling effects are severely restricted by the additional constraint arising from the integrability condition.