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

Although it is well known that the Ward identities prohibit anomalous dimensions for conserved currents in local field theories, a claim from certain holographic models involving bulk dilaton couplings is that the gauge field associated with the boundary current can acquire an anomalous dimension. We resolve this conundrum by showing that all the bulk actions that produce anomalous dimensions for the conserved current generate non-local actions at the boundary. In particular, the Maxwell equations are fractional. To prove this, we generalize to p-forms the Caffarelli/Silvestre (CS) extension theorem. In the context of scalar fields, this theorem demonstrates that second-order elliptic differential equations in the upper half-plane in \({\mathbb{R}_{+}^{n+1}}\) reduce to one with the fractional Laplacian, \({\Delta^{\gamma}}\), with \({\gamma \in {\mathbb{R}}}\), when one of the dimensions is eliminated. From the p-form generalization of the CS extension theorem, we show that at the boundary of the relevant holographic models, a fractional gauge theory emerges with equations of motion of the form, \({\Delta^{\gamma} A^{t} = 0}\) with \({\gamma \in R}\) and At the boundary components of 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.