  We introduce a novel geometrically frustrated classical Ising model, dubbed
the "spin vorticity model", whose ground state manifold is a novel classical
spin liquid, a "2-form Coulomb phase". We study the thermodynamics of this
model both analytically and numerically, exposing the presence of algebraically
decaying correlations and demonstrating an extensive ground state entropy, and
give a comprehensive account of its ground state properties and excitations.
Each classical ground state may be decomposed into collections of closed
2-dimensional membranes, supporting fractionalized string excitations attached
to the boundaries of open membranes. At finite temperature, the model can then
be described as a gas of closed strings in a background of fluctuating
membranes. We demonstrate that the emergent gauge structure of the
low-temperature phase is naturally captured in the formalism of 2-form
electrodynamics, which describes 1-dimensional charged strings coupled to a
rank-2 anti-symmetric gauge field. After establishing the classical spin
vorticity model, we consider perturbing it with quantum exchange interactions,
from which we derive an effective membrane exchange model of the quantum
dynamics of these membranes, which maps to a frustrated 2-form U(1) lattice
gauge theory. We show the existence of a fine-tuned Rokhsar-Kivelson point
where the quantum ground state is an equal weight superposition of all
classical ground state configurations. We further demonstrate how to quantize
the string excitations, by coupling a 1-form string field to the emergent
2-form U(1) gauge field, thus mapping a quantum spin model to a 2-form
gauge-Higgs model. We discuss the stability of the gapless deconfined phase of
this gauge theory and the possibility of realizing a novel class of phases of
quantum matter: 2-form U(1) quantum spin liquids.
