  A recent work (arXiv:1811.04930) proposed a SU(2) gauge theory for optimal
doping criticality in the cuprate superconductors. The theory contains $N_h$
Higgs fields transforming under the adjoint representation of SU(2), with
$N_h=1$ for the electron-doped cuprates, and $N_h=4$ for the hole-doped
cuprates. We investigate the strong-coupling dynamics of this gauge theory,
while ignoring the coupling to fermionic excitations. We integrate out the
SU(2) gauge field in a strong-coupling expansion, and obtain a lattice action
for the Higgs fields alone. We study such a lattice action, with O($N_h$)
global symmetry, in an analytic large $N_h$ expansion and by Monte Carlo
simulations for $N_h=4$ and find consistent results. We find a confining phase
with O($N_h$) symmetry preserved (this describes the Fermi liquid phase in the
cuprates), and Higgs phases (describing the pseudogap phase of the cuprates)
with different patterns of the broken global O($N_h$) symmetry. One of the
Higgs phases is topologically trivial, implying the absence of any excitations
with residual gauge charges. The other Higgs phase has $\mathbb{Z}_2$
topological order, with `vison' excitations carrying a $\mathbb{Z}_2$ gauge
charge. We find consistent regimes of stability for the topological Higgs phase
in both our numerical and analytical analyses.
