Topological charges in 2D $\mathcal{N}=(2,2)$ theories and massive BPS states

We study how charges of global symmetries that are manifest in the ultraviolet definition of a theory are realized as topological charges in its infrared 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 Bogomol’nyi-Prasad-Sommerfield (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 four-dimensional $\mathcal{N}=2$ 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 four-dimensional 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.