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We study how charges of global symmetries that are manifest in theultra-violet definition of a theory are realized as topological charges in itsinfra-red effective theory for two-dimensional theories with$\mathcal{N}=(2,2)$ supersymmetry. We focus on the charges that the statesliving on $S^1$ carry. The central charge---or BPS masses---of thesupersymmetry algebra play a crucial role in making this correspondenceprecise. We study two examples: $U(1)$ gauge theories with chiral matter, andworld-volume theories of "dynamical surface operators" of 4d $\mathcal{N}=2$gauge theories. In the former example, we show that the flavor charges of thetheory are realized as topological winding numbers in the effective theory onthe Coulomb branch. In the latter, we show that there is a one-to-onecorrespondence between topological charges of the effective theory of thedynamical surface operator and the electric, magnetic, and flavor charges ofthe 4d gauge theory. We also examine the topologically charged massive BPSstates on $S^1$ and discover that the massive BPS spectrum is sensitive to theradius of the circle in the simplest theory---the free theory of a periodictwisted chiral field. We clarify this behavior by showing that the massive BPSspectrum on $S^1$, unlike the BPS ground states, cannot be identified aselements of a cohomology.

Topological charges in 2d N=(2,2) theories and massive BPS states