Winding Number in String Field Theory

TL;DR

This paper explores the concept of a winding number in cubic string field theory, proposing a regularization method to assign quantized values and analyzing the implications for solutions like the tachyon vacuum.

Contribution

It introduces a regularization approach to define a winding number in CSFT and examines its properties and limitations for different string configurations.

Findings

Regularization via shifting K yields quantized winding numbers for the tachyon vacuum.

The additive law for winding numbers does not hold for certain configurations.

Some string field configurations do not satisfy the equations of motion strongly, affecting their interpretation.

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 N=(\pi^2/3)\int(U Q_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=\int 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 KBc 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)^{\pm 1}]=\mp 1 for U_tv corresponding to the tachyon vacuum. However, we find that N[(U_tv)^{\pm 2}] differs from \mp 2, the value expected from the additive law of N. This result may be understood from the fact that \Psi=U Q_B U^{-1} with U=(U_tv)^{\pm 2} does not satisfy the CSFT EOM in the strong sense and hence is not truly a pure-gauge in our regularization.