  We show that the stochastic field theory for directed percolation in presence
of an additional conservation law (the C-DP class) can be mapped exactly to the
continuum theory for the depinning of an elastic interface in short-range
correlated quenched disorder. On one line of parameters commonly studied, this
mapping leads to the simplest overdamped dynamics. Away from this line, an
additional memory term arises in the interface dynamics; we argue that it does
not change the universality class. Since C-DP is believed to describe the Manna
class of self-organized criticality, this shows that Manna stochastic sandpiles
and disordered elastic interfaces (i.e. the quenched Edwards-Wilkinson model)
share the same universal large-scale behavior.
