Star Spectroscopy in the Constant B field Background

TL;DR

This paper analyzes the spectrum of Neumann matrices with zero modes in Witten's cubic string field theory under a constant B field, revealing spectral properties and implications for brane tension and star algebra factorization.

Contribution

It provides a detailed spectral analysis of Neumann matrices with zero modes in a B field, including continuous and discrete spectra, and explores implications for string field theory and star algebra.

Findings

Continuous spectrum within [-1/3, 0)

Degeneracy patterns of eigenvectors in the spectrum

Proof that brane tension ratio is one

Abstract

In this paper we calculate the spectrum of Neumann matrix with zero modes in the presence of the constant B field in Witten's cubic string field theory. We find both the continuous spectrum inside $[{-1\over3}, 0)$ and the constraint on the existence of the discrete spectrum. For generic $\theta$, -1/3 is not in the discrete spectrum but in the continuous spectrum. For each eigenvalue in the continuous spectrum there are four twist-definite degenerate eigenvector except for -1/3 at which the degeneracy is two. However, for each twist-definite eigenvector the twist parity is opposite among the two spacetime components. Based upon the result at -1/3 we prove that the ratio of brane tension to be one as expected. Furthermore, we discuss the factorization of star algebra in the presence of B field under zero-slope limit and comment on the implications of our results to the recent proposed map of Witten's star to Moyal's star.