Winding number for arbitrary integer value in Cubic String Field Theory

We have focused on the topological structure of Cubic string field theory (CSFT). From the similarity of action between CSFT and Chern-Simons (CS) theory in three dimensions, we have investigated the quantity ${\cal N}=\pi^2/3\int (UQU^{-1})^3$, which is expected to be the counterpart of winding number in CS theory. In our previous research, it was reported that $\mathcal{N}$ can only take a limited number of integer values due to the inevitable anomalies in Okawa type solution. To overcome this unsatisfactory results, we evaluate $\mathcal{N}$ and EOM against a solution itself, $\mathcal{T}$, for more general class of pure gauge form solution written in $K,B$ and $c$ in this paper. Then we obtain general formula of $\mathcal{N}$ and $\mathcal{T}$. From this result, we show that there is an infinite number of solutions that $\mathcal{N}$ takes any integer value while keeping $\cal T=0$. We also show the gauge invariant observable of these solutions take appropriate values. Furthermore, we evaluate the integral form of the BRST-exact quantity as surface integral.