Generalised anomalies, QCD$_4$, and holography

During the last decade, the notion of an 't Hooft anomaly has been generalised to the case of discrete symmetries. An interesting instance, discussed by Tanizaki, is the mixed anomaly between the discrete axial symmetry and the flavour and baryonic symmetries in massless QCD$_4$. The goal of this note is to provide a derivation of this anomaly from a top-down holographic dual of QCD$_4$. It is found that the topological couplings in the bulk supergravity dual of the D4-D8 system encode Tanizaki's anomaly, once fluctuations around the bulk gauge fields are turned on. A technical challenge for this computation is the difficulty in maintaining gauge invariance of supergravity theories in the presence of D-branes. To overcome this issue, a compact formulation of the flux sector of (massive) type IIA supergravity in the presence of D8 branes is presented. A crucial ingredient in the analysis is the need for smearing the D8 branes, in order to impose $\tau$-shift invariance on the fluctuation ansatz.