  Quantum topological excitations (skyrmions) are analyzed from the point of
view of their duality to spin excitations in the different phases of a
disordered two-dimensional, short-range interacting, SO(3) quantum magnetic
system of Heisenberg type. The phase diagram displays all the phases, which are
allowed by the duality relation. We study the large distance behavior of the
two-point correlation function of quantum skyrmions in each of these phases
and, out of this, extract information about the energy spectrum and
non-triviality of these excitations. The skyrmion correlators present a
power-law decay in the spin-glass(SG)-phase, indicating that these quantum
topological excitations are gapless but nontrivial in this phase. The SG phase
is dual to the AF phase, in the sense that topological and spin excitations are
respectively gapless in each of them. The Berezinskii-Kosterlitz-Thouless
mechanism guarantees the survival of the SG phase at $T \neq 0$, whereas the AF
phase is washed out to T=0 by the quantum fluctuations. Our results suggest a
new, more symmetric way of characterizing a SG-phase: one for which both the
order and disorder parameters vanish, namely $<\sigma > = 0 $, $<\mu > =0 $,
where $\sigma$ is the spin and $\mu$ is the topological excitation operators.
