  We report measurements of the photon beam asymmetry $\Sigma$ for the
reactions $\vec{\gamma}p\to p\pi^0$ and $\vec{\gamma}p\to p\eta $ from the
GlueX experiment using a 9 GeV linearly-polarized, tagged photon beam incident
on a liquid hydrogen target in Jefferson Lab's Hall D. The asymmetries,
measured as a function of the proton momentum transfer, possess greater
precision than previous $\pi^0$ measurements and are the first $\eta$
measurements in this energy regime. The results are compared with theoretical
predictions based on $t$-channel, quasi-particle exchange and constrain the
axial-vector component of the neutral meson production mechanism in these
models.
