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We study 4D systems in which parameters of the theory have positiondependence in one spatial direction. In the limit where these parameters jump,this can lead to 3D interfaces supporting localized degrees of freedom. Apriori, this sort of position dependence can occur at either weak or strongcoupling. Demanding time-reversal invariance for $U(1)$ gauge theories with aduality group $\Gamma \subset SL(2,\mathbb{Z})$ leads to interfaces at strongcoupling which are characterized by the real component of a modular curvespecified by $\Gamma$. This provides a geometric method for extracting theelectric and magnetic charges of possible localized states. We illustrate thesegeneral considerations by analyzing some 4D $\mathcal{N} = 2$ theories with 3Dinterfaces. These 4D systems can also be interpreted as descending from asix-dimensional theory compactified on a three-manifold generated by a familyof Riemann surfaces fibered over the real line. We show more generally that 6Dsuperconformal field theories compactified on such spaces also produce trappedmatter by using the known structure of anomalies in the resulting 4D bulktheories.

Geometric Approach to 3D Interfaces at Strong Coupling