Anomaly Inflow and Holography

We systematically study the perturbative anomaly inflow by the bulk Chern-Simons (CS) theory in five-dimensional anti-de Sitter spacetime ($\text{AdS}_5$). When the bulk geometry is chosen to be AdS, along with the standard bulk-boundary interplay, an additional, holographic dual description emerges. Introduction of UV and IR 3-branes makes the anomaly story remarkably rich and many interesting aspects can be obtained. With Neumann boundary conditions (BC) on the IR brane, the dual CFT has an unbroken symmetry, which then can either be weakly gauged by choosing Neumann UV-BC, or kept as a purely global symmetry with Dirichlet UV-BC. This corresponds to the holographic realization of `t Hooft anomaly matching, either for ABJ or `t Hooft anomalies, or both. On the other hand, when the IR-BC breaks the bulk gauge group $G$ down to a subgroup $H_1$, the dual 4D CFT has a spontaneously broken symmetry. In this case, we describe how the (gauged) Wess-Zumino-Witten action emerges naturally from the bulk CS action. In particular, we discuss that, unlike in the case of Neumann IR-BC where 5D gauge invariance is restored by IR brane-localized fermions, with $G/H_1$ IR-BC no localized modes are required. Nevertheless, anomaly matching is fulfilled by delocalized modes, namely Wilson lines along the fifth dimension, and these are Goldstone bosons (GB) in the dual 4D theory. When some part of $G$ is weakly gauged we show that, thanks to a proper field redefinition of the corresponding source fields, the "would-be" GBs can be completely removed, consistently with our standard expectation. We demonstrate how the most general case, a typical situation occurring in models of dynamical symmetry breaking, may be analyzed with our formalism. Finally, we discuss the quantization condition of the CS level, both with Neumann and Dirichlet BC.