  We develop a gauge theory of the critical behavior of the topological
excitations-driven Berezinskii-Kosterlitz-Thouless (BKT) phase transition in
the XY model with weak quenched disorder. We find that while in two-dimensions
the liquid of topological defects exhibits the BKT critical behavior, the
three-dimensional system shows more singular Vogel-Fulcher-Tamman criticality
heralding its freezing into a spin glass. Our findings provide insights into
the topological origin of spin glass formation.
