  We evaluate axial vector transition form factors in holographic QCD models
that have been shown to reproduce well recent experimental and theoretical
results for the pion transition form factor. Comparing with L3 data on
$f_1\to\gamma\gamma^*$ we find remarkable agreement regarding the shape of
single-virtual form factors. In the double-virtual case, the holographic
results differ strongly from a simple dipole form, and this has an important
impact on the corresponding estimate of the axial vector contribution to the
anomalous magnetic moment of the muon $a_\mu$ through hadronic light-by-light
scattering. We demonstrate that hard-wall models satisfy the
Melnikov-Vainshtein short-distance constraint for the latter, if and only if
the infinite tower of axial vector states is included. The results for $a_\mu$,
however, are strongly dominated by the first few resonances. Numerically, these
results turn out to be surprisingly large: (2.9 - 4.1)$\times 10^{-10}$ in the
hard-wall models, 57-58% of which are due to the longitudinal contribution,
which is the one responsible for the Melnikov-Vainshtein short-distance
constraint. Rescaling the holographic result to obtain an optimal fit of L3
data, but then matching only 52% of the asymptotic constraint, the result is
reduced to $2.2(5)\times 10^{-10}$, which is still significantly larger than
most previous phenomenological estimates of the axial vector exchange
contribution.
