We investigate the transformation from ordinary gauge field to noncommutative one which was introduced by N. . It is shown that the general transformation which is determined only by gauge equivalence has a path dependence in 'θ-space'. This ambiguity is negligible when we compare the ordinary Dirac-Born-Infeld action with the noncommutative one in the U(1) case, because of the U(1) nature and slowly varying field approximation. However, in general, in the higher derivative approximation or in the U(N) case, the ambiguity cannot be neglected due to its noncommutative structure. This ambiguity corresponds to the degrees of freedom of field re-Gauge theories on noncommutative spaces have been investigated for many years from mathematical and physical viewpoint ([Connes] and the references in [SW]). Especially in string theory, the worldvolume theory of D-branes in a background B-field is described by noncommutative Yang-Mills or Dirac-Born-Infeld theory.Recently, Seiberg and Witten [SW] argued the equivalence between ordinary gauge field theory and the noncommutative one as the low energy effective theories of open strings: they arise from the same two-dimensional field theory regularlized in different ways, so that there must be a transformation among them. In [SW], this transformation is uniquely given by the gauge equivalence relation, and this implies the equivalence between ordinary Dirac-Born-Infeld action and the noncommutative one.In this short note, we re-examine the validity of above arguments and point out that the transformation of [SW] has in general ambiguities. In section 2, we begin with the gauge equivalence relation between two nearby points in the 'θ-space' and show that there is ambiguity with arbitrary constant parameters. Then, we discuss the path dependence of the transformation in the 'θ-space', which is found by considering the commutator of two transformations. This implies the existence of another ambiguity. In section 3 we investigate these ambiguities from different viewpoint. Next in section 4, we consider the U(1) case in slowly varying field approximation. This is the situation of [SW] in comparing the ordinary Dirac-Born-Infeld action with the noncommutative one. In this case the ambiguities do not affect the result of [SW], because of the U(1) nature and of neglecting the higher derivative terms. In section 5, we summarize the paper and give some discussions. In "Note Added", we argue that the path dependence can be reduced to the field redefinition.

In this paper, we study a new matrix theory based on non-BPS D-instantons in type IIA string theory and D-instanton -anti D-instanton system in type IIB string theory, which we call K-matrix theory. The theory correctly incorporates the creation and annihilation processes of D-branes. The configurations of the theory are identified with spectral triples, which are the noncommutative generalization of Riemannian geometryá la Connes, and they represent the geometry on the world-volume of higher dimensional D-branes. Remarkably, the configurations of D-branes in the K-matrix theory are naturally classified by a K-theoretical version of homology group, called Khomology. Furthermore, we argue that the K-homology correctly classifies the D-brane configurations from a geometrical point of view. We also construct the boundary states corresponding to the configurations of the K-matrix theory, and explicitly show that they represent the higher dimensional D-branes. This Fredholm module (H (0) , H (1) , φ 0 , φ 1 , F ) describes the configurations of the IIB K-matrix theory in an analogous way as above. H (0) and H (1) corresponds to the Chan-Paton indices of D-instantons and anti D-instantons, respectively. The *-homomorphisms φ 0 and φ 1 is used to obtain the configurations that D-instantons and anti D-instantons are settled inside the space-time manifold in the same way as explained in section 3.3 for non-BPS D-instantons. F is again the normalized tachyon field T b . Then, (4.4) and (4.5) are the conditions corresponding to (2.8) and (2.9), respectively.

We examine the fluctuations around a Dp-brane solution in an unstable D-brane system using boundary states and also boundary string field theory. We show that the fluctuations correctly reproduce the fields on the Dp-brane. Plugging these into the action of the unstable D-brane system, we recover not only the tension and RR charge, but also full effective action of the Dp-brane exactly. Our method works for general unstable D-brane systems and provides a simple proof of D-brane descent/ascent relations under the tachyon condensation. In the lowest dimensional unstable D-brane system, called K-matrix theory, D-branes are described in terms of operator algebra. We show the equivalence of the geometric and algebraic descriptions of a D-brane world-volume manifold using the equivalence between path integral and operator formulation of the boundary quantum mechanics. As a corollary, the Atiyah-Singer index theorem is naturally obtained by looking at the coupling to RR-fields. We also generalize the argument to type I string theory.

We analyse unstable D-brane systems in type I string theory. Generalizing the proposal in hep-th/0108085, we give a physical interpretation for real KK-theory and claim that the D-branes embedded in a product space X ×Y which are made from the unstable Dpbrane system wrapped on Y are classified by a real KK-theory group KKO p−1 (X, Y ). The field contents of the unstable D-brane systems are systematically described by a hidden Clifford algebra structure.We also investigate the matrix theory based on non-BPS D-instantons and show that the spectrum of D-branes in the theory is exactly what we expect in type I string theory, including stable non-BPS D-branes with Z 2 charge. We explicitly construct the D-brane solutions in the framework of BSFT and analyse the physical property making use of the Clifford algebra. type I string theory. We are particularly interested in the construction of D-branes in these systems. As shown in [4,2,12], type I string theory has non-trivial D-brane spectrum even when the space-time manifold is flat. In fact, the charge of flat Dpbranes transverse to R 9−p is classified by the real K-theory group KO(R 9−p ) 4 [2], which is given in table 1. In particular, there are stable non-BPS D-branes with Z 2 p −1 0 1 2 3 4 5 6 7 8 9 KO(R 9−p ) Z 2 Z 2 Z 0 0 0 Z 0 Z 2 Z 2 Z Table 1: The spectrum of flat Dp-branes in type I theory, which is classified by the real K-theory group KO(R 9−p ).charges. So, it would be more challenging to explore this theory than the type II sting theory. We will show that we correctly obtain this spectrum using type I K-matrix theory.It is easy to generalize the idea to higher dimensional systems. Actually, we will not restrict our arguments to the matrix theory, but consider D-branes made by unstable Dp-brane systems in type I string theory, (i.e. non-BPS Dp-branes for p = −1, 0, 2, 3, 4, 6, 7, 8 and Dp-brane -anti Dp-brane system for p = 1, 5, 9). The gauge groups and the representation of tachyon fields on the world-volume of non-BPS Dp-branes are examined in [14], and the results are summarized in table 2. 5 It is also shown in [14, 2] that the K-theory group which classifies the charge of D-branes made by the descent relations from the unstable Dp-brane system is the real K-theory groupwhere Y is the world-volume manifold of the system.From the argument analogous to that given in type II case, it is quite natural to expect that the classification of type I D-branes via the real K-theory is generalized by using real KK-theory, when we take into account the D-branes stretched along the directions transverse to the unstable Dp-brane system. As we will explain in section 4 Here KO(R n ) denotes the reduced K-theory group of S n , KO(S n ) [13]. 5 In this table, we omitted the massless scalar and tachyon modes from Dp-D9 strings for p ≥ 5, since they do not affect the K-theory classification, as discussed in [14,15]. So, we simply neglect them in this paper.

We investigate the Hopf algebra structure in string worldsheet theory and give a unified formulation of the quantization of string and the space-time symmetry. We reformulate the path integral quantization of string as a Drinfeld twist at the worldsheet level. The coboundary relation shows that the Drinfeld twist defines a module algebra which is equivalent to operators with normal ordering. Upon applying the twist, the space-time diffeomorphism is deformed into a twisted Hopf algebra, while the Poincar\'e symmetry is unchanged. This suggests a characterization of the symmetry: unbroken symmetries are twist invariant Hopf subalgebras, while broken symmetries are realized as twisted ones. We provide arguments that relate this twisted Hopf algebra to symmetries in path integral quantization.Comment: 35 pages, no figure, v2: references and comments added, typos corrected, v3: requires PTP style, title changed, final version published in PT