  We study the non-equilibrium evolution of coexisting ferromagnetic domains in
the two-dimensional quantum Ising model -- a setup relevant in several
contexts, from quantum nucleation dynamics and false-vacuum decay scenarios to
recent experiments with Rydberg-atom arrays. We demonstrate that the
quantum-fluctuating interface delimiting a large bubble can be studied as an
effective one-dimensional system through a "holographic" mapping. For the
considered model, the emergent interface excitations map to an integrable chain
of fermionic particles. We discuss how this integrability is broken by
geometric features of the bubbles and by corrections in inverse powers of the
ferromagnetic coupling, and provide a lower bound to the timescale after which
the bubble is ultimately expected to melt. Remarkably, we demonstrate that a
symmetry-breaking longitudinal field gives rise to a robust ergodicity breaking
in two dimensions, a phenomenon underpinned by Stark many-body localization of
the emergent fermionic excitations of the interface.
