Winding number for arbitrary integer value in Cubic String Field Theory

Abstract: 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 $\cal N$ 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 $\cal N$ and EOM against a solution itself, $\cal T$, for more general class of pure gauge form solution written in $K,B$ and $c$ in this paper. Then we obtain general formula of $\cal N$ and $\cal T$. From this result, we show that there is an infinite number of solutions that $\cal N$ 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.