We study four-dimensional (4D) systems in which parameters of the theory have position dependence in one spatial direction. In the limit where these parameters jump, this can lead to 3D interfaces supporting localized degrees of freedom. A?priori , this sort of position dependence can occur at either weak or strong coupling. Demanding time-reversal invariance for U ( 1 ) gauge theories with a duality group Γ ? S L ( 2 , Z ) leads to interfaces at strong coupling which are characterized by the real component of a modular curve specified by Γ . This provides a geometric method for extracting the electric and magnetic charges of possible localized states. We illustrate these general considerations by analyzing some 4D N = 2 theories with 3D interfaces. These 4D systems can also be interpreted as descending from a six-dimensional theory compactified on a three-manifold generated by a family of Riemann surfaces fibered over the real line. We show more generally that 6D superconformal field theories compactified on such spaces also produce trapped matter by using the known structure of anomalies in the resulting 4D bulk theories.
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机译：我们研究了四维（4D）系统，其中理论的参数具有在一个空间方向上的位置依赖性。在这些参数跳跃的极限中，这可以导致支持局部自由度的3D接口。 a？先验，可以在弱或强耦合方面发生这种位置依赖性。对u（1）尺寸γ的尺寸γ响起的时间反转不变性？ S L（2，Z）导致强耦合的接口，其特征在于由γ指定的模块化曲线的真实成分。这提供了一种用于提取可能局部状态的电磁性电荷的几何方法。我们通过使用3D接口分析一些4D n = 2理论来说明这些一般注意事项。这些4D系统也可以解释为从六维理论的下降，该六维理论在由Riemann表面系在真实线上的riemann表面系列产生的三个歧管上进行压实。我们展示了更多一般地说，在这些空间上压缩的6D超成形场理论也通过使用所得的4D散装理论中的已知异常结构产生截留的物质。
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