The singular geometry of the sliver

TL;DR

This work analyzes sliver states in Witten's cubic open string field theory as projection operators in the matter star algebra, revealing their intrinsic singular geometry. It demonstrates a zero-slope factorization ${\cal A}\to{\cal A}_0\otimes{\cal A}_1$, where ${\cal A}_0$ captures spacetime structure and ${\cal A}_1$ encodes higher string modes, and shows that slivers correspond to D-brane-like projections and can exhibit a split-string picture with midpoint constraints. The authors derive analytic eigenvectors with eigenvalue $-1$ for the sliver matrices, linking these to geometric constraints such as midpoint localization and half-string splitting, and interpret these singularities in terms of skyscraper-like sheaf structures. They further argue that closed strings can be represented within open string field theory by special squeezed states satisfying midpoint relations, suggesting a framework in which open-string diagrams cover closed-string moduli and hinting at a deep connection between open and closed string descriptions. The discussion also addresses mathematical pathologies, potential regularization, and broad generalizations (e.g., including a $B$-field and level truncation as regulators), highlighting the role of singular states in understanding D-branes and the emergence of closed strings in the open-string framework.

Abstract

We consider "sliver" states which act as projection operators in the matter star product of Witten's cubic string field theory. These sliver states, which might be associated with Dirichlet p-branes, are not finite norm states in the matter string Hilbert space. We describe the singularities of these states, and demonstrate that the sliver states are composed of strings having singular geometric features. These singularities take a particularly simple form in the zero slope limit alpha' -> 0, where the star algebra factorizes into a product of the algebra of functions on space-time and the noncommutative star product of fields associated with higher string modes. An analogy to the sliver geometry suggests a natural mechanism for describing closed string states in open string field theory.