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We show that the stochastic field theory for directed percolation in presenceof an additional conservation law (the C-DP class) can be mapped exactly to thecontinuum theory for the depinning of an elastic interface in short-rangecorrelated quenched disorder. On one line of parameters commonly studied, thismapping leads to the simplest overdamped dynamics. Away from this line, anadditional memory term arises in the interface dynamics; we argue that it doesnot change the universality class. Since C-DP is believed to describe the Mannaclass of self-organized criticality, this shows that Manna stochastic sandpilesand disordered elastic interfaces (i.e. the quenched Edwards-Wilkinson model)share the same universal large-scale behavior.

An exact mapping of the stochastic field theory for Manna sandpiles to interfaces in random media