BCFT and Sliver state

We give a comment on the possible rôle of the sliver state in the generic boundary conformal field theory. We argue that for each Cardy state, there exists at least one projector in the string field theory. ∗E-mail address: [email protected] One of the long standing problem in the string theory is the description of the geometry. Generically we do not need the space-time at all to define the consistent string background. What is necessary instead is the modular invariant representation of the (super) Virasoro algebra with the certain central extension. If we need the geometry, we have to extract it in principle from the algebraic data of the CFT, namely the set of the primary fields, their OPE coefficients, the modular invariant combination of the left and right movers and so on. In the development of the noncommutative geometry [1, 2], a hint to this problem is given in the context of the noncommutative soliton[3]. In this approach the description of the space-time is replaced by the (in general noncommutative) C-algebra A. The noncommutative soliton is defined as the projection operator of A. Immediately after the discovery, it is interpreted as the D-brane in the presence of the background B field [4]. In the noncommutative geometry, a canonical way to extract the geometry from the algebraic data is through the study their K-group [5, 6] which are classified into two types, K0(A) and K1(A). The latter is identified as the isomorphic class of the unitary operator in MatN×N(A) for large enough N . On the other hand the former one is represented by the projection operator in MatN×N(A) which fits naturally with the very definition of the noncommutative soliton. In the commutative geometry, the D-brane charge is classified by the topological K-group [7, 8]. The noncommutative soliton gives the representation of K-homology group of the operator algebra and thus describes the D-brane in the noncommutative situation[9, 10]. Since the current examples of the noncommutative geometry is restricted to the rather simple spaces such as the Moyal plane or the noncommutative torus, it is desirable to extend such framework to the full string theory. In this language the operator algebra A should be replaced by the full (boundary) CFT module. This is, actually, the philosophy advocated by Witten long ago [11] (see also [12]) in his open string field theory. Recently there is a remarkable progress toward this direction [13, 14, 15, 16, 17, 18]. It is based on the conjecture that we may find the ghost string field which satisfies, QΨgh = Ψgh ⋆Ψgh (1) and if one factories the full string field as Ψ = Ψm ⊗ Ψgh, the equation of motion for the matter part becomes, Ψm ⋆Ψm = Ψm , (2)