  We study an effective field theory of a vortex lattice in a two-dimensional
neutral rotating superfluid. Utilizing particle-vortex dualities, we explore
its formulation in terms of a $U(1)$ gauge theory coupled to elasticity, that
at low energies reduces to a compact Lifshitz theory augmented with a Berry
phase term encoding the vortex dynamics in the presence of a superflow.
Utilizing elasticity- and Lifshitz-gauge theory dualities, we derive dual
formulations of the vortex lattice in terms of a traceless symmetric
scalar-charge theory and demonstrate low-energy equivalence of our dual gauge
theory to its elasticity-gauge theory dual. We further discuss a multipole
symmetry of the vortex lattice and its dual gauge theory's multipole one-form
symmetries. We also study its topological crystalline defects, where the
multipole one-form symmetry plays a prominent role. It classifies the defects,
explains their restricted mobility, and characterizes descendant vortex phases,
which includes a novel vortex supersolid phase. Using the dual gauge theory, we
also develop a mean-field theory for the quantum melting transition from a
vortex crystal to a vortex supersolid.
