Type-IV 't Hooft Anomalies on the Lattice: Emergent Higher-Categorical Symmetries and Applications to LSM Systems

't Hooft anomalies impose fundamental constraints on quantum matter and often lead to emergent symmetry structures upon gauging. We analyze a lattice model with four global symmetries realizing a mixed anomaly described by sim a_1wedge a_2wedge a_3wedge a₄, where the a_i denote background gauge fields for the global symmetries. Through explicit lattice gauging, we demonstrate the emergence of higher symmetry structures, including 2-group, non-invertible, and higher fusion categorical symmetries. We also provide a field-theoretical understanding of these results. Applying this framework to systems with Lieb-Schultz-Mattis anomalies, obtained by promoting part of the internal symmetries to translational symmetries, we demonstrate that modulated (dipole) symmetries arise as direct counterparts of those in systems with purely internal typeIV anomalies. Importantly, we uncover a qualitatively new feature absent in previously studied modulated symmetries: their realization can become intrinsically defect-dependent. In particular, the emergent symmetry structure changes depending on whether symmetry defects are present. This work establishes a concrete lattice realization of mixed anomalies and reveals a rich structure of emergent symmetries, thereby clarifying their role in constraining quantum phases of matter.