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Open String States around a Classical Solution in Vacuum String Field Theory

Hiroyuki Hata, Teruhiko Kawano

TL;DR

This work constructs an exact classical solution $\Psi_c$ in Vacuum String Field Theory in the Siegel gauge and analyzes whether it represents the perturbative open string vacuum. By enforcing BRST invariance, the pure-ghost operator ${\cal Q}$ becomes fixed, and fluctuations around $\Psi_c$ yield a tachyon with mass squared $m_t^2$ approaching $-1$ and a massless vector mode, though full transversality is not established. The potential height is studied via the energy density ${\cal E}_c$ and the D25-brane tension $T_{25}$, with the ratio ${\cal E}_c/T_{25}$ expressed in terms of Neumann coefficients; however, level-truncation and determinant ambiguities lead to unexpected, unsettled results, signaling unresolved issues in the height problem. The paper highlights both promising signs—tachyon and vector fluctuations—and substantial hurdles needing further clarification, including the complete fluctuation spectrum and a robust treatment of determinant factors in the energy ratio. Overall, it provides important insights into VSFT’s connection to perturbative open string theory while identifying key questions for future work.

Abstract

We construct a classical solution of vacuum string field theory (VSFT) and study whether it represents the perturbative open string vacuum. Our solution is given as a squeezed state in the Siegel gauge, and it fixes the arbitrary coefficients in the BRST operator in VSFT. We identify the tachyon and massless vector states as fluctuation modes around the classical solution. The tachyon mass squared αm_t^2 is given in a closed form using the Neumann coefficients defining the three-string vertex, and it reproduces numerically the expected value of -1 to high precision. The ratio of the potential height of the solution to the D25-brane tension is also given in terms of the Neumann coefficients. However, the behavior of the potential height in level truncation does not match our expectation, though there are subtle points in the analysis.

Open String States around a Classical Solution in Vacuum String Field Theory

TL;DR

This work constructs an exact classical solution in Vacuum String Field Theory in the Siegel gauge and analyzes whether it represents the perturbative open string vacuum. By enforcing BRST invariance, the pure-ghost operator becomes fixed, and fluctuations around yield a tachyon with mass squared approaching and a massless vector mode, though full transversality is not established. The potential height is studied via the energy density and the D25-brane tension , with the ratio expressed in terms of Neumann coefficients; however, level-truncation and determinant ambiguities lead to unexpected, unsettled results, signaling unresolved issues in the height problem. The paper highlights both promising signs—tachyon and vector fluctuations—and substantial hurdles needing further clarification, including the complete fluctuation spectrum and a robust treatment of determinant factors in the energy ratio. Overall, it provides important insights into VSFT’s connection to perturbative open string theory while identifying key questions for future work.

Abstract

We construct a classical solution of vacuum string field theory (VSFT) and study whether it represents the perturbative open string vacuum. Our solution is given as a squeezed state in the Siegel gauge, and it fixes the arbitrary coefficients in the BRST operator in VSFT. We identify the tachyon and massless vector states as fluctuation modes around the classical solution. The tachyon mass squared αm_t^2 is given in a closed form using the Neumann coefficients defining the three-string vertex, and it reproduces numerically the expected value of -1 to high precision. The ratio of the potential height of the solution to the D25-brane tension is also given in terms of the Neumann coefficients. However, the behavior of the potential height in level truncation does not match our expectation, though there are subtle points in the analysis.

Paper Structure

This paper contains 18 sections, 125 equations, 3 tables.

Table of Contents

  1. Introduction
  2. VSFT action
  3. Classical solution in the Siegel gauge
  4. Fluctuation spectrum around $\bm{\Psi_{\rm c}}$
  5. Tachyon wave function
  6. Determination of $\bm{t}$
  7. Tachyon mass
  8. Massless vector mode
  9. Potential height
  10. Energy density $\bm{{\cal E}_{\rm c}}$
  11. Normalization factor $\bm{{\cal N}_{\rm t}}$
  12. $\bm{g_{\rm o}}$ as three-tachyon on-shell coupling
  13. $\bm{{\cal E}_{\rm c}/T_{25}}$
  14. Reexamination of $\bm{{\cal E}_{\rm c}/T_{25}}$
  15. Conclusion
  16. ...and 3 more sections