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

Berezinskii-Kosterlitz-Thouless and Vogel-Fulcher-Tammann criticality in $\mathrm{XY}$ model