Ghost story. III. Back to ghost number zero

After having defined a 3-strings midpoint-inserted vertex for the bc system, we analyze the relation between gh=0 states (wedge states) and gh=3 midpoint duals. We find explicit and regular relations connecting the two objects. In the case of wedge states this allows us to write down a spectral decomposition for the gh=0 Neumann matrices, despite the fact that they are not commuting with the matrix representation of K1. We thus trace back the origin of this noncommutativity to be a consequence of the imaginary poles of the wedge eigenvalues in the complex k-plane. With explicit reconstruction formulas at hand for both gh=0 and gh=3, we can finally show how the midpoint vertex avoids this intrinsic noncommutativity at gh=0, making everything as simple as the zero momentum matter sector.Comment: 40 pages. v2: typos and minor corrections, presentation improved in sect. 4.3, plots added in app. A.1, two refs added. To appear in JHE

A fresh look at midpoint singularities in the algebra of string fields

In this paper we study the midpoint structure of the algebra of open strings from the standpoint of the operator/Moyal formalism. We construct a split string description for the continuous Moyal product of hep-th/0202087, study the breakdown of associativity in the star algebra, and identify in infinite sequence of new (anti)commutative coordinates for the star product in in the complex plane. We also explain how poles in the open string non(anti)commutativity parameter correspond to certain ``null'' operators which annihilate the vertex, implying that states proportional to such operators tend to have vanishing star product with other string fields. The existence of such poles, we argue, presents an obstruction to realizing a well-defined formulation of the theory in terms of a Moyal product. We also comment on the interesting, but singular, representation $L_0$ which has appeared prominently in the recent studies of Bars {\it et al}.Comment: 40 pages, 5 figures. Version to be submitted to JHEP. Some interesting and previouusly unpublished results are included here. These include both an interpretation of poles in the open string noncommutativity parameter as corresponding to null operators in the algebra, and an identification of an infinite sequence of new commutative and null coordinates in the complex $\kappa$ plan

Solving Witten's SFT by Insertion of Operators on Projectors

Following Okawa, we insert operators at the boundary of regulated star algebra projectors to construct the leading order tachyon vacuum solution of open string field theory. We also calculate the energy density of the solution and the ratio between the kinetic and the cubic terms. A universal relationship between these two quantities is found. We show that for any twist invariant projector, the energy density can account for at most 68.46% of the D25-brane tension. The general results are then applied to regulated slivers and butterflies, and the next-to-leading order solution for regulated sliver states is constructed.Comment: 24 pages, 4 figure

Geometry of discrete-time spin systems

Classical Hamiltonian spin systems are continuous dynamical systems on the symplectic phase space $(S^2)^n$. In this paper we investigate the underlying geometry of a time discretization scheme for classical Hamiltonian spin systems called the spherical midpoint method. As it turns out, this method displays a range of interesting geometrical features, that yield insights and sets out general strategies for geometric time discretizations of Hamiltonian systems on non-canonical symplectic manifolds. In particular, our study provides two new, completely geometric proofs that the discrete-time spin systems obtained by the spherical midpoint method preserve symplecticity. The study follows two paths. First, we introduce an extended version of the Hopf fibration to show that the spherical midpoint method can be seen as originating from the classical midpoint method on $T^*\mathbf{R}^{2n}$ for a collective Hamiltonian. Symplecticity is then a direct, geometric consequence. Second, we propose a new discretization scheme on Riemannian manifolds called the Riemannian midpoint method. We determine its properties with respect to isometries and Riemannian submersions and, as a special case, we show that the spherical midpoint method is of this type for a non-Euclidean metric. In combination with K\"ahler geometry, this provides another geometric proof of symplecticity.Comment: 17 pages, 2 figures. arXiv admin note: substantial text overlap with arXiv:1402.333

Open String Fields As Matrices

We present a new representation of the string vertices of the cubic open string field theory. By using this three-string vertex, we attempt to identify open string fields as huge-sized matrices by following Witten's idea. By using these huge matrices, we obtain some results about the construction of partial isometries in the algebra of open string fields.Comment: 24 pages, lanlmac; (v2) references added; (v3) typos corrected and one reference adde