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Star Algebra Spectroscopy

Leonardo Rastelli, Ashoke Sen, Barton Zwiebach

TL;DR

The paper provides a complete analytic diagonalization of the star-product Neumann matrices in OSFT by exploiting the derivation $K_1=L_1+L_{-1}$, revealing a continuous spectrum for $M$ on the interval $[-\tfrac{1}{3},0)$ with twist-degenerate eigenvectors except at $-\tfrac{1}{3}$. It shows that eigenvectors of ${\rm K_1}$ commute with $M$, $M^{12}$, and $M^{21}$, enabling closed-form expressions for their eigenvalues in terms of a continuous parameter $\kappa$, e.g. $\mu(\kappa)=-\frac{1}{1+2\cosh(\pi\kappa/2)}$, and similarly for $M^{12}$ and $M^{21}$. A simpler matrix $B$ is introduced to relate the eigenvalues of $T$ (wedge states) to those of $M$, leading to a consistent functional interpretation of the spectrum and a rich wedge/sliver structure. The work further develops the spectral density and finite-level truncation analysis, providing explicit density formulas and numerical checks that support the continuous spectrum picture and offer a framework for computing determinants and tensions of D-branes within OSFT. Overall, the results illuminate the spectral landscape governing the star algebra and wedge states, with implications for nonperturbative formulations of string field theory.

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.

Star Algebra Spectroscopy

TL;DR

The paper provides a complete analytic diagonalization of the star-product Neumann matrices in OSFT by exploiting the derivation , revealing a continuous spectrum for on the interval with twist-degenerate eigenvectors except at . It shows that eigenvectors of commute with , , and , enabling closed-form expressions for their eigenvalues in terms of a continuous parameter , e.g. , and similarly for and . A simpler matrix is introduced to relate the eigenvalues of (wedge states) to those of , leading to a consistent functional interpretation of the spectrum and a rich wedge/sliver structure. The work further develops the spectral density and finite-level truncation analysis, providing explicit density formulas and numerical checks that support the continuous spectrum picture and offer a framework for computing determinants and tensions of D-branes within OSFT. Overall, the results illuminate the spectral landscape governing the star algebra and wedge states, with implications for nonperturbative formulations of string field theory.

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.

Paper Structure

This paper contains 14 sections, 123 equations, 2 figures.

Table of Contents

  1. Introduction and Summary
  2. Notation and Definitions
  3. The $-1/3$ Eigenvector of $M$
  4. ${\rm K_1}$ and its Eigenvectors
  5. Wedge States, ${\rm K_1}$ and $M$
  6. Eigenvectors of $T$ and $M$
  7. Relating the eigenvalues of $M$ and ${\rm K_1}$ via the $B$ matrices
  8. Diagonalization of $\rho_1$, $\rho_2$, $M^{12}$ and $M^{21}$
  9. String Functionals and the Spectrum
  10. Spectral Density and Finite Level Analysis
  11. Quantization condition on the eigenvalues for finite matrices
  12. Eigenvalue distribution function
  13. Numerical tests of the spectrum
  14. Open questions

Figures (2)

  • Figure 1: This figure shows two plots of eigenvalues $\mu_n$ of the level truncated matrix $M_L$. In one of them $L=64$, and in the other $L=128$. On the horizontal axis we have $n$ referring to the $n$-th eigenvalue with $n=1$ labelling the smallest eigenvalue (closest to $-1/3$). On the vertical axis we show $\ln(-\mu_n)$. Note that the eigenvalues become small very fast. The solid line shows the predicted curve for $L=128$, ignoring corrections of order $1/\ln L$.
  • Figure 2: This figure shows two plots of the positive eigenvalues $\kappa_n$ of the level truncated matrix ${\rm K_1}$. In one of them $L=64$, and in the other $L=128$. On the horizontal axis we have $n$ referring to the $n$-th eigenvalue with $n=1$ labelling the smallest eigenvalue. On the vertical axis we show $\kappa_n$. The solid line shows the predicted answer for $L=128$ ignoring corrections of order $1/\ln L$.