Motivated by the similarity between cubic string field theory (CSFT) and the Chern-Simons theory in three dimensions, we study the possibility of interpreting N = (π 2 /3) (UQ B U −1 ) 3 as a kind of winding number in CSFT taking quantized values. In particular, we focus on the expression of N as the integration of a BRST-exact quantity, N = Q B A, which vanishes identically in naive treatments. For realizing non-trivial N , we need a regularization for divergences from the zero eigenvalue of the operator K in the KB c algebra. This regularization must at same time violate the BRST-exactness of the integrand of N . By adopting the regularization of shifting K by a positive infinitesimal, we obtain the desired value N [(U tv ) ±1 ] = ∓1 for U tv corresponding to the tachyon vacuum. However, we find that N [(U tv ) ±2 ] differs from ∓2, the value expected from the additive law of N . This result may be understood from the fact that Ψ = UQ B U −1 with U = (U tv ) ±2 does not satisfy the CSFT EOM in the strong sense and hence is not truly a pure-gauge in our regularization.

In a previous paper [arXiv:1111.2389], we studied the multi-brane solutions in cubic string field theory by focusing on the topological nature of the "winding number" N which counts the number of branes. We found that N can be nontrivial owing to the singularity from the zero-eigenvalue of K of the KBc algebra, and that solutions carrying integer N and satisfying the EOM in the strong sense is possible only for N = 0, ±1. In this paper, we extend the construction of multibrane solutions to |N | ≥ 2. The solutions with N = ±2 is made possible by the fact that the correlator is invariant under a transformation exchanging K with 1/K and hence K = ∞ eigenvalue plays the same role as K = 0. We further propose a method of constructing solutions with |N | ≥ 3 by expressing the eigenvalue space of K as a sum of intervals where the construction for |N | ≤ 2 is applicable.

We show how conformal field theory topological defects can relate solutions of open string field theory for different boundary conditions. To this end we generalize the results of Graham and Watts to include the action of defects on boundary condition changing fields. Special care is devoted to the general case when nontrivial multiplicities arise upon defect action. Surprisingly the fusion algebra of defects is realized on open string fields only up to a (star algebra) isomorphism.