Star Algebra Spectroscopy

TL;DR

This paper analytically determines the eigenvalues and eigenvectors of key Neumann matrices in string field theory, revealing their spectral properties and implications for star algebra states.

Contribution

It provides exact analytic expressions for the spectra of Neumann matrices using the derivation K_1, advancing understanding of the star algebra structure.

Findings

Spectrum of M^{11} is continuous in [-1/3, 0)

Eigenvectors are degenerate twist even and odd

Eigenvalue -1/3 is exceptional

Abstract

The spectrum of the infinite dimensional Neumann matrices M^{11}, M^{12} and M^{21} in the oscillator construction of the three-string vertex determines key properties of the star product and of wedge and sliver states. We study the spectrum of eigenvalues and eigenvectors of these matrices using the derivation K_1 = L_1 + L_{-1} of the star algebra, which defines a simple infinite matrix commuting with the Neumann matrices. By an exact calculation of the spectrum of K_1, and by consideration of an operator generating wedge states, we are able to find analytic expressions for the eigenvalues and eigenvectors of the Neumann matrices and for the spectral density. The spectrum of M^{11} is continuous in the range [-1/3, 0) with degenerate twist even and twist odd eigenvectors for every eigenvalue except for -1/3.