A Note on D - Branes of Odd Codimensions

On a noncommutative space of rank-1, we construct a codimension-one soliton explicitly and, in the context of noncommutative bosonic open string field theory, identify it with the D24-brane. We compute the tension of the proposed D24-brane, yielding an exact value and show that it is related to the tension of the co-dimension two D23-brane by the string T-duality. This resolves a puzzle posed by the result of Harvey, Kraus, Larsen and Martinec. We also apply the result to non-BPS D-branes in superstring theories and argue that the codimension-one soliton gives rise to new descent relations among the non-BPS D-branes in Type IIA and Type IIB string theories via T-duality. Work supported in part by BK-21 Initiative in Physics (SNU Project 2), KRF International Collaboration Grant, and KOSEF Interdisciplinary Research Grant 98-07-02-07-01-5. In a series of recent important developments [1, 2, 3], it has been noted that noncommutative field theories in various dimensions arise quite naturally at various corners of the moduli space of nonperturbative string theories, especially when a nonzero value of the NS two-form potential B2 is turned on, inducing noncommutativity on the open string dynamics. Among the most interesting developments are noncommutative solitons, especially in the context of the string theories. At infinite noncommutativity limit, Gopakumar, Minwalla and Strominger [4] have constructed explicit forms of static solitons by utilizing an isomorphism between fields on noncommutative plane and operators on a single-particle Hilbert space, Hθ —viz. the Weyl-Moyal correspondence. The method of [4] were adapted by Harvey, Kraus, Larsen and Martinec [7] and Dasgupta, Mukhi and Rajesh [6] to bosonic string and superstring theories, respectively, and have constructed a noncommutative version of D-branes, supporting the viewpoint that D-branes are built out of solitonic lumps of tachyon living on higher dimensional D-branes [9]. The result of [7, 6] has an important implication for Sen’s conjecture [8] concerning universality of the tachyon potential in that, in the limit of infinitely strong noncommutativity the level-zero truncation yields exactly Sen’s conjectural relation between the D-brane tension TD and the height of the tachyon potential V (T ): TD + ( V (0)− V (T0) ) = 0. In Ref. [7], in the infinite noncommutativity limit, bosonic D-branes were constructed explicitly only for those of even codimensions, viz. D23-, D21-, ... D1-branes but the rest, viz. D24-, D22-, ..... D0-branes, were left out. In the commutative limit, all Dp-branes are on equal footing, as they are constructed out of the tachyon field as localized energy lumps of both odd and even co-dimensions. It is highly unlikely that, for some mysterious reasons, the D-branes of odd codimensions are suppressed relative to those of even codimensions as noncommutativity is turned on. How then do the D-branes of odd codimensions arise in the infinite noncommutativity limit? In this Letter, we point out that, in both bosonic string and superstring theories, the D-branes of odd codimensions can be constructed by applying a sequence of U(∞) symmetry transformations. We will begin with posing an issue concerning noncommutative Dp-branes in bosonic open string theory. In doing so, we will bring up the puzzle concerning missing Dp-branes for p=even more explicit. At level-zero truncation, the bosonic open string field theory is described by the action of the tachyon field T : S0 = C Gst ∫ dy √ −G ( 1 2 GF (T )∂μT∂νT + · · · − V (T ) )