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We relate moduli stabilization ($V'=0$) in the bulk of $AdS_D$ or $dS_D$ tobasic properties of the Wilsonian effective action in the holographic dualtheory on $dS_{D-1}$: the single-trace terms in the action have vanishing betafunctions, and higher-trace couplings are determined purely from lower-traceones. In the de Sitter case, this encodes the maximal symmetry of the bulkspacetime in a quantity which is accessible within an observer patch. Along theway, we clarify the role of counterterms, constraints, and operator redundancyin the Wilsonian holographic RG prescription, reproducing the expected behaviorof 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 energycorresponds to a nonperturbatively small imaginary contribution to theWilsonian action of pure de Sitter, a result consistent with the need foradditional degrees of freedom in the holographic description of its ultimatedecay.

Moduli Stabilization and the Holographic RG for AdS and dS