Winding number in string field theory

Abstract

Motivated by the similarity between cubic string field theory (CSFT) and the Chern-Simons theory in three dimensions, we study the possibility of interpreting $\mathcal{N} = \left( {{\pi^2}/3} \right)\int {{{\left( {U{\mathcal{Q}_B}{U^{{ - 1}}}} \right)}^3}}$ as a kind of winding number in CSFT taking quantized values. In particular, we focus on the expression of $\mathcal{N}$ as the integration of a BRST-exact quantity, $\mathcal{N} = \int {{\mathcal{Q}_B}\mathcal{A}}$ which vanishes identically in naive treatments. For realizing non-trivial $\mathcal{N}$, we need a regularization for divergences from the zero eigenvalue of the operator K in the KB c algebra. This regularization must at the same time violate the BRST-exactness of the integrand of $\mathcal{N}$. By adopting the regularization of shifting K by a positive infinitesimal, we obtain the desired value $\mathcal{N}[{({{\text{U}}_{\text{tv}}})^{{\pm {1}}}}] = \mp {1}$ for U _tv corresponding to the tachyon vacuum. However, we find that $\mathcal{N}[{({{\text{U}}_{\text{tv}}})^{{\pm {2}}}}]$ differs from ∓2, the value expected from the additive law of $\mathcal{N}$. This result may be understood from the fact that $\Psi = U{\mathcal{Q}_B}{U^{{ - 1}}}$ 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.