  Topological states of matter are at the root of some of the most fascinating
phenomena in condensed matter physics. Here we argue that skyrmions in the
pseudo-spin space related to an emerging SU(2) symmetry enlighten many
mysterious properties of the pseudogap phase in under-doped cuprates. We detail
the role of the SU(2) symmetry in controlling the phase diagram of the
cuprates, in particular how a cascade of phase transitions explains the arising
of the pseudogap, superconducting and charge modulation phases seen at low
temperature. We specify the structure of the charge modulations inside the
vortex core below $T_{c}$, as well as in a wide temperature region above
$T_{c}$, which is a signature of the skyrmion topological structure. We argue
that the underlying SU(2) symmetry is the main structure controlling the
emergent complexity of excitations at the pseudogap scale $T^{*}$. The theory
yields a gapping of a large part of the anti-nodal region of the Brillouin
zone, along with $q=0$ phase transitions, of both nematic and loop currents
characters.
