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Although it is well known that the Ward identities prohibit anomalousdimensions for conserved currents in local field theories, a claim from certainholographic models involving bulk dilaton couplings is that the gauge fieldassociated with the boundary current can acquire an anomalous dimension. Weresolve this conundrum by showing that all the bulk actions that produceanomalous dimensions for the conserved current generate non-local actions atthe boundary. In particular, the Maxwell equations are fractional. To provethis, we generalize to p-forms the Caffarelli/Silvestre (CS) extension theorem.In the context of scalar fields, this theorem demonstrates that second-orderelliptic differential equations in the upper half-plane in ${\mathbbR}_+^{n+1}$ reduce to one with the fractional Laplacian, $\Delta^\gamma$, whenone of the dimensions is eliminated. From the p-form generalization of the CSextension theorem, we show that at the boundary of the relevant holographicmodels, a fractional gauge theory emerges with equations of motion of the form,$\Delta ^\gamma A^t= 0$ with $\gamma$ $\in R$ and $A^t$ the boundary componentsof the gauge field. The corresponding field strength $F = d_\gamma A^t= d\Delta^\frac{\gamma-1}{2} A^t$ is invariant under $A^t \rightarrow A^t+ d_\gamma\Lambda$ with the fractional differential given by  $d_\gamma \equiv (\Delta)^\frac{\gamma-1}{2}d$, implying that $[A^t]=\gamma$which is in general not unity.

Anomalous Dimensions for Boundary Conserved Currents in Holography via the Caffarelli-Silvestri Mechanism for p-forms