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We investigate the interplay of generalized global symmetries in 2+1 dimensions by introducing a lattice model that couples a ZN clock model to a ZN gauge theory via a topological interaction. This coupling binds the charges of one symmetry to the disorder operators of the other, and when these composite objects condense, they give rise to emergent generalized symmetries with mixed ’t Hooft anomalies. These anomalies result in phases with ordinary symmetry breaking, topological order, and symmetry-protected topological (SPT) order, where the different types of order are not independent but intimately related. We further explore the gapped boundary states of these exotic phases and develop theories for phase transitions between them. Additionally, we extend our lattice model to incorporate a non-invertible global symmetry, which can be spontaneously broken, leading to domain walls with non-trivial fusion rules. 

A long-standing problem in the study of topological phases of matter has been to understand the types of fractional topological insulator (FTI) phases possible in 3+1 dimensions. Unlike ordinary topological insulators of free fermions, FTI phases are characterized by fractional 𝜃-angles,long-range entanglement, and fractionalization. Starting from a simple family of ℤ_N lattice gauge theories due to Cardy and Rabinovici, we develop a class of FTI phases based on the physical mechanism of oblique confinement and the modern language of generalized global symmetries. We dub these phases oblique topological insulators. Oblique TIs arise when dyons—bound states of electric charges and monopoles—condense, leading to FTI phases characterized by topological order, emergent one-forms symmetries, and gapped boundary states not realizable in 2+1-D alone.Based on the lattice gauge theory, we present continuum topological quantum field theories (TQFTs) for oblique TI phases involving fluctuating one-form and two-form gauge fields. We show explicitly that these TQFTs capture both the generalized global symmetries and topological orders seen in the lattice gauge theory. We also demonstrate that these theories exhibit a universal “generalized magneto-electric effect” in the presence of two-form background gauge fields. Moreover,we characterize the possible boundary topological orders of oblique TIs,finding a new set of boundary states not studied previously for these kinds of TQFTs.