  We study torsional oscillations of neutron stars in the scalar-tensor theory
of gravity using the relativistic Cowling approximation. We compute unperturbed
neutron star models adopting realistic equations of state for the neutron
star's core and crust. For scalar-tensor theories that allow for spontaneous
scalarization, the crust thickness can be significantly smaller than in general
relativity. We derive the perturbation equation describing torsional
oscillations in scalar-tensor theory, and we solve the corresponding eigenvalue
problem to find the oscillation frequencies. The fundamental mode (overtone)
frequencies become smaller (larger) than in general relativity for scalarized
stellar models. Torsional oscillation frequencies may yield information on the
crust microphysics if microphysics effects are not degenerate with
strong-gravity effects, such as those due to scalarization. To address this
issue, we consider two different models for the equation of state of the crust
and we look at the effects of electron screening. The effect of scalarization
on torsional oscillation frequencies turns out to be smaller than uncertainties
in the microphysics for all spontaneous scalarization models allowed by binary
pulsar observations. Our study shows that the observation of quasi-periodic
oscillations (QPOs) following giant flares can be used to constrain the
microphysics of neutron star crusts, whether spontaneous scalarization occurs
or not.
