  We consider the vector and the pseudovector glueball in the top-down
holographic model of large-$N_c$ QCD of Witten and their decays into ordinary
mesons described by the D8 brane construction due to Sakai and Sugimoto. At
leading order, the relevant interactions are determined exclusively by the
Chern-Simons action of the D8 branes and are thus rigidly connected to the
chiral anomaly and the Wess-Zumino-Witten terms. As found in a previous study
of the pseudovector glueball, which we revisit and complete, the resulting
decay widths are surprisingly large, implying that both the pseudovector and
the vector glueball are very broad resonances, with a conspicuous dominance of
decays into $a_1\rho$ and $K_1(1400) K^*$ in the case of the vector glueball.
We also obtain a certain weak mixing of vector glueballs with ordinary vector
mesons, but we conclude that it does not provide an explanation for the
so-called $\rho\pi$ puzzle in charmonium decays.
