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

Duality, Quantum Skyrmions and the Stability of an SO(3) Two-Dimensional Quantum Spin-Glass