Star Algebra Projectors

TL;DR

The paper identifies a broad class of open-string star-algebra projectors realized as surface states whose open-string midpoint lies on the boundary, and shows that their wave-functionals split into left and right halves while remaining invariant under opposite translations of half-strings. By analyzing degenerate conformal maps and pinching limits, it demonstrates that these projectors arise naturally from conformal field theory on degenerate disks and that their half-string projections are themselves surface states. Beyond the canonical sliver, it constructs and analyzes the butterfly and a family of generalized butterflies, including the nothing state, and proves projector properties, universal Neumann-matrix eigenvectors, and explicit oscillator representations. The work also extends the construction to BCFTs, explores related simple Virasoro-projectors, and discusses potential implications for vacuum string field theory and the algebraic understanding of the star product, including connections to continuous Moyal-type decompositions. Overall, it provides a geometric mechanism for a large class of rank-one projectors and deepens the link between conformal maps, half-string formalisms, and OSFT solutions.

Abstract

Surface states are open string field configurations which arise from Riemann surfaces with a boundary and form a subalgebra of the star algebra. We find that a general class of star algebra projectors arise from surface states where the open string midpoint reaches the boundary of the surface. The projector property of the state and the split nature of its wave-functional arise because of a nontrivial feature of conformal maps of nearly degenerate surfaces. Moreover, all such projectors are invariant under constant and opposite translations of their half-strings. We show that the half-string states associated to these projectors are themselves surface states. In addition to the sliver, we identify other interesting projectors. These include a butterfly state, which is the tensor product of half-string vacua, and a nothing state, where the Riemann surface collapses.

Figures (23)

• Figure 1: The generic kind of surface state providing a projector of the star algebra. The open string is the vertical boundary, and the open string midpoint is shown with a heavy dot. The rest of the boundary has open string boundary condition. Note that this part of the boundary touches the open string midpoint.
• Figure 2: The geometry involved in computing the inner product of a surface state $|\Sigma\rangle$ with itself.
• Figure 3: The geometry involved in computing the star product of a surface state with itself. The local coordinate patch, shown as the shaded half-disk to the right, is to be glued to the shaded region of the diagram representing $\widehat{{\cal S}}$.
• Figure 4: The surface $R(t)$ is pinching for $t=1$. The pinching point $P$ separates the regions $R_1$ and $R_2$ of the surface $R(1)$.
• Figure 5: Illustration of a conformal map from the upper-half plane plus a strip of width $\pi$ connected by a narrow neck (part (a)) to the upper-half plane (part (b)).