  We show that the boundary states satisfy a nonlinear relation (the
idempotency equation) with respect to the star product of closed string field
theory. This relation is universal in the sense that various D-branes,
including the infinitesimally deformed ones, satisfy the same equation,
including the coefficient. This paper generalizes our analysis (hep-th/0306189)
in the following senses. (1) We present a background-independent formulation
based on conformal field theory. It illuminates the geometric nature of the
relation and allows us to more systematically analyze the variations around the
D-brane background. (2) We show that the Witten-type star product satisfies a
similar relation but with a more divergent coefficient. (3) We determine the
coefficient of the relation analytically. The result shows that the alpha
parameter can be formally factored out, and the relation becomes universal. We
present a conjecture on vacuum theory based on this computation.
