Theory of oblique topological insulators

A long-standing problem in the study of topological phases of matter has been to under-stand the types of fractional topological insulator (FTI) phases possible in 3+1 dimen-sions. Unlike ordinary topological insulators of free fermions, FTI phases are charac-terized by fractional Theta-angles, long-range entanglement, and fractionalization. Starting from a simple family of ZN 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 topo-logical insulators. Oblique TIs arise when dyons-bound states of electric charges and monopoles-condense, leading to FTI phases characterized by topological order, emer-gent one-form symmetries, and gapped boundary states not realizable in 2+1-D alone. Based on the lattice gauge theory, we present continuum topological quantum field the-ories (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 symme-tries and topological orders seen in the lattice gauge theory. We also demonstrate that these theories exhibit a universal "generalized magnetoelectric effect" in the presence of two-form background gauge fields. Moreover, we characterize the possible bound-ary topological orders of oblique TIs, finding a new set of boundary states not studied previously for these kinds of TQFTs.