  Motivated from mathematical aspects of the superstring theory, we introduce a
new equation on a balanced, hermitian manifold, with zero first Chern class.
Solving the equation, one will obtain, in each Bott--Chern cohomology class, a
balanced metric which is hermitian Ricci--flat. This can be viewed as a
differential form level generalization of the classical Calabi--Yau equation.
We establish the existence and uniqueness of the equation on complex tori, and
prove certain uniqueness and openness on a general K\"ahler manifold.
