usual renormalization group equations. The numerical coefficient $b=8a$ is $b=20$ for ${\cal N}=8$ supergravity, $b=8(1 + n_V/4)$ for ${\cal N}=4$ supergravity with $n_V$ matter multiplets, and similarly for other examples.

Our computations are all made in bulk and there is no reference to a boundary theory. This is a rather old fashioned point of view but it is worthwhile for interpreting the set up as a toy model for the physical cosmological constant. For this we imagine the signs such that the cosmological constant is positive and the running such that it becomes small at large distances. The dimensionless coupling $e^2$ would be tuned to take a tiny value, of order $10^{-120}$. The absence of power law corrections would then ensure naturalness in the sense that the logarithmic running is so mild that quantum corrections would preserve the enormous hierarchy. This mechanism does not explain the smallness of the observed cosmological constant but it offers a viable scenario for its naturalness.

Quantum Inequivalence and Boundary Modes.
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In this section we discuss the interplay between the trace of the energy momentum tensor and quantum inequivalence between duality frames. We interpret quantum inequivalence as a physical effect due to boundary modes. 