  We study 3+1 dimensional $SU(N)$ Quantum Chromodynamics (QCD) with $N_f$
degenerate quarks that have a spatially varying complex mass. It leads to a
network of interfaces connected by interface junctions. We use anomaly inflow
to constrain these defects. Based on the chiral Lagrangian and the conjectures
on the interfaces, characterized by a spatially varying $\theta$-parameter, we
propose a low-energy description of such networks of interfaces. Interestingly,
we observe that the operators in the effective field theories on the junctions
can carry baryon charges, and their spin and isospin representations coincide
with baryons. We also study defects, characterized by spatially varying
coupling constants, in 2+1 dimensional Chern-Simons-matter theories and in a
3+1 dimensional real scalar theory.
