in part by these and related developments. Computations in these contexts share the techniques we employ and offer some confidence in their applicability.

One Loop Quantum Corrections in AdS$_4$
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In this section we employ heat kernel methods to compute the one loop contributions to the anomalous trace of the energy momentum tensor in AdS$_4$ from fields with various spins. We interpret the resulting divergences in the effective action as logarithmic running of the effective cosmological constant.

Notation and Review
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One loop quantum corrections in Euclidean quantum gravity are determined by a Gaussian path integral with the schematic form, $$W=-\ln \int\mathcal{D}\phi e^{-\phi \Box \phi} = \frac{1}{2}\ln \det \Box = \frac{1}{2}\sum_i \ln \lambda_i~,$$ where the $\phi$ denotes the collection of linearized fields, $\Box$ represents their kinetic operator, and $\lambda_i$ are the eigenvalues of $\Box$. We represent the effective action $W$ in terms of the heat kernel $D(t) = \sum_i e^{-t\lambda_i}$ as $$\label{eq:Wdef}
W = -\int_{\epsilon^2}^{\infty}\frac{dt}{2t} D(t)~,$$ where $\epsilon$ is a UV regulator with dimension of length. It is customary to express results for heat kernels in terms of the (equal point) heat kernel density $K(t)$ expanded at small $t$ $$\label{eq:Kdef}
K(t) =  \frac{1}{\textrm{Vol}_{\textrm{AdS}_4}} D(t)=  \frac{1}{(4\pi t)^2}\bigg( 1 + a_2 t 