What is the generalization, if any, of the weak and dominant energy conditions to SUGRA?

In standard general relativity, we have the null energy condition, the weak energy condition related to stability, and the dominant energy condition related to forbidding superluminal causal influences. With a cosmological constant added, the standard lore is to subtract the cosmological constant contribution first.

However, strange things happen in anti de Sitter space with a negative cosmological constant. It can easily be shown that for an asymptotically AdS space, it's possible to include tachyonic scalers, as long as their mass squared isn't lower than $-ck^2$ where k is the radius of curvature and c is a precise coefficient which depends upon the precise units chosen. Such a model will still be stable, and the ADM energy still turns out to be nonnegative! Interestingly enough, this lower bound is satisfied naturally in SUGRA theories. Clearly, the weak energy condition is violated, but this doesn't rule out stability.

Another example. Suppose a SUGRA theory has one or more chiral superfields and admits two or more stable supersymmetric phases. If the superpotential is W, they are characterized by $\partial W =0$. In SUGRA, the cosmological constant goes as $-|W|^2$. Suppose phase I has a smaller $|W|$. Then, it might appear at first glance that it is metastable, and will eventually decay to phase II after an exponentially long time. Certainly, with respect to phase I, WEC is violated. That's not so. Magically, the domain wall tension and hyperbolic geometry come to the rescue. The domain wall tension goes at least as $c\left[ |W_2|^2 - |W_1|^2 \right]$ as obtained from BPS analysis. Once again, the coefficient c in SUGRA is such that the total ADM energy still goes up. In fact, c saturates the bound. It differs from the flat spacetime case where the energy goes down by $R^3$ where R is the bubble radius and the tension contribution only as $R^2$ so that for large enough R, the total energy will go down. Because of hyperbolic geometry, asymptotically, both contributions scale exponentially in R.

Naturally, the question is what is the proper generalization of WEC to SUGRA for purposes of proving stability, and the generalization of DEC for causality?

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Both examples are flawed. The second fails because it is just not true that bulk and boundary go as $R^3$ and $R^2$ in AdS, the hyperbolic geometry make both go as an exponential in R asymptotically, and therefore they are asymptotically equal up to a constant factor. This is why you have a minimum boundary/bulk ratio, and why the vacuum is stable in the first place.