The Spectrum of the Neumann Matrix with Zero Modes

TL;DR

This paper analyzes the spectrum of the Neumann matrix with zero modes in string field theory, revealing a new eigenvalue and eigenvectors, and identifying a critical parameter affecting their form.

Contribution

It provides an analytical calculation of the spectrum of the Neumann matrix including zero modes, uncovering new eigenvalues and eigenvectors, and introduces a critical parameter influencing their structure.

Findings

Discovered an additional eigenvalue in (0,1) range.

Derived explicit eigenvectors and their generating functions.

Identified a critical parameter b_0 = 8 ln 2 affecting eigenvector forms.

Abstract

We calculate the spectrum of the matrix M' of Neumann coefficients of the Witten vertex, expressed in the oscillator basis including the zero-mode a_0. We find that in addition to the known continuous spectrum inside [-1/3,0) of the matrix M without the zero-modes, there is also an additional eigenvalue inside (0,1). For every eigenvalue, there is a pair of eigenvectors, a twist-even and a twist-odd. We give analytically these eigenvectors as well as the generating function for their components. Also, we have found an interesting critical parameter b_0 = 8 ln 2 on which the forms of the eigenvectors depend.