Higher-form Gauge Symmetries in Multipole Topological Phases

In this article we study field-theoretical aspects of multipolar topological insulators. Previous research has shown that such systems naturally couple to higher-rank tensor gauge fields that arise as a result of gauging dipole or subsystem $U(1)$ symmetries. Here we propose a complementary framework using electric higher-form symmetries. We utilize the fact that gauging 1-form electric symmetries results in a 2-form gauge field which couples naturally to extended line-like objects: Wilson lines. In our context the Wilson lines are electric flux lines associated to the electric polarization of the system. This allows us to define a generalized 2-form Peierls' substitution for dipoles that shows that the off-diagonal components of a rank-2 tensor gauge field $A_{ij}$ can arise as a lattice Peierls factor generated by the background antisymmetric 2-form gauge field. This framework has immediate applications: (i) it allows us to construct a manifestly topological quadrupolar response action given by a Dixmier-Douady invariant -- a generalization of a Chern number for 2-form gauge fields -- which makes plain the quantization of the quadrupole moment in the presence of certain crystal symmetries; (ii) it allows for a clearer interpretation of the rank-2 Berry phase calculation of the quadrupole moment; (iii) it allows for a proof of a generic Lieb-Schultz-Mattis theorem for dipole-conserving systems.