  Imaging atmospheric Cherenkov telescopes (IACTs) that are sensitive to
potential $\gamma$-ray signals from dark matter (DM) annihilation above
$\sim50$ GeV will soon be superseded by the Cherenkov Telescope Array (CTA).
CTA will have a point source sensitivity an order of magnitude better than
currently operating IACTs and will cover a broad energy range between 20 GeV
and 300 TeV. Using effective field theory and simplified models to calculate
$\gamma$-ray spectra resulting from DM annihilation, we compare the prospects
to constrain such models with CTA observations of the Galactic center with
current and near-future measurements at the Large Hadron Collider (LHC) and
direct detection experiments. For DM annihilations via vector or pseudoscalar
couplings, CTA observations will be able to probe DM models out of reach of the
LHC, and, if DM is coupled to standard fermions by a pseudoscalar particle,
beyond the limits of current direct detection experiments.
