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

TL;DR

This paper explores how global symmetry charges in 2d N=(2,2) theories are represented as topological charges in the IR, revealing their relation to BPS states and the impact of circle radius on the spectrum.

Contribution

It demonstrates the realization of flavor and gauge charges as topological winding numbers and clarifies the behavior of massive BPS states on S^1 in 2d N=(2,2) theories.

Findings

Flavor charges are topological winding numbers on the Coulomb branch.

Topological charges correspond to electric, magnetic, and flavor charges in 4d theories.

Massive BPS spectrum on S^1 depends on circle radius and cannot be described by cohomology.

Abstract

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.