  We study the metastable dynamics of a discretised version of the
mass-conserving stochastic Allen-Cahn equation. Consider a periodic
one-dimensional lattice with $N$ sites, and attach to each site a real-valued
variable, which can be interpreted as a spin, as the concentration of one type
of metal in an alloy, or as a particle density. Each of these variables is
subjected to a local force deriving from a symmetric double-well potential, to
a weak ferromagnetic coupling with its nearest neighbours, and to independent
white noise. In addition, the dynamics is constrained to have constant total
magnetisation or mass. Using tools from the theory of metastable diffusion
processes, we show that the long-term dynamics of this system is similar to a
Kawasaki-type exchange dynamics, and determine explicit expressions for its
transition probabilities. This allows us to describe the system in terms of the
dynamics of its interfaces, and to compute an Eyring-Kramers formula for its
spectral gap. In particular, we obtain that the spectral gap scales like the
inverse system size squared.
