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

High Energy Physics - Theory