A Note on the Path Integral Representation of the Boundary State of D-brane

Abstract: In this paper we construct path integral representations of the boundary states in some special backgrounds such as the U(1) gauge field background, the linear dilaton background and the open string tachyon background. The initial purpose of this paper is to construct a general solution of the boundary conformal field theory with the analytical approach, mainly for the constraint equations(Ln− L̃−n)|B〉 = 0 are difficult to be solved to obtain the solution represented by string modes from the pure algebraic approach. However in the path integral representation it is easy transforming those algebraic equations into the differential equations which can be solved. Another purpose of this paper is to try to explore an open question. we do not know how to construct an exact theory of D-branes in the general background until now. However many recent researches show the boundary state description indeed seizes some fundamental features of D-branes in the rather special backgrounds. Since the general background field effects can be easily introduced in the path integral representation, we argue that path integral representation of the boundary state should provide an exact description of D branes in the general backgrounds.