Moduli Stabilization and the Holographic RG for AdS and dS

We relate moduli stabilization ($V'=0$) in the bulk of $AdS_D$ or $dS_D$ to basic properties of the Wilsonian effective action in the holographic dual theory on $dS_{D-1}$: the single-trace terms in the action have vanishing beta functions, and higher-trace couplings are determined purely from lower-trace ones. In the de Sitter case, this encodes the maximal symmetry of the bulk spacetime in a quantity which is accessible within an observer patch. Along the way, we clarify the role of counterterms, constraints, and operator redundancy in the Wilsonian holographic RG prescription, reproducing the expected behavior of the trace of the stress-energy tensor in the dual for both $AdS_D$ and $dS_D$. We further show that metastability of the gravity-side potential energy corresponds to a nonperturbatively small imaginary contribution to the Wilsonian action of pure de Sitter, a result consistent with the need for additional degrees of freedom in the holographic description of its ultimate decay.

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