Physical States at the Tachyonic Vacuum of Open String Field Theory

TL;DR

This paper develops a gauge-fixed, level-truncated OSFT framework around the tachyon vacuum to count physical open-string states via Fadeev-Popov determinants, testing Sen's conjecture that ghost-number-0 cohomology vanishes. It derives a nonperturbative long exact sequence relating absolute and relative BRST cohomologies and uses numerical data up to level 9 to show FP index vanishes for the observed multiplet while negative ghost-number cohomologies H^(−1) and H^(−2) are nonempty with dimension 1. The results confirm the absence of open-string excitations at the tachyon vacuum in the FP sense but reveal a richer nonperturbative BRST cohomology structure, including nontrivial negative ghost-number sectors. These findings refine our understanding of the nonperturbative BRST algebra in OSFT and lay groundwork for extensions to higher-spin sectors and improved numerical analyses.

Abstract

We illustrate a method for computing the number of physical states of open string theory at the stable tachyonic vacuum in level truncation approximation. The method is based on the analysis of the gauge-fixed open string field theory quadratic action that includes Fadeev-Popov ghost string fields. Computations up to level 9 in the scalar sector are consistent with Sen's conjecture about the absence of physical open string states at the tachyonic vacuum. We also derive a long exact cohomology sequence that relates relative and absolute cohomologies of the BRS operator at the non-perturbative vacuum. We use this exact result in conjunction with our numerical findings to conclude that the higher ghost number non-perturbative BRS cohomologies are non-empty.

Figures (8)

• Figure 1: The first group of zeros of $\Delta^{ (n)}_{-}(p^2)$, for $n=0,-1,-2$ at levels $L= 4,\ldots,9$.
• Figure 2: Zeros of $\Delta^{ (n)}_{-}(p^2)$ (a) and of $\Delta^{ (n)}_{+}(p^2)$ (b) for $n=0,-1,-2$ at levels $L= 4,\ldots,9$ up to $p^2 =-10$.
• Figure 3: The vanishing eigenvalues of $C^{ (n,-)}_L(p)$ for $n=0,-1,-2$ and $p^2\approx -2$ at level L=9.
• Figure 4: Eigenvalues of $K^{ (n)}(p)$ (a) and of level truncated $K^{ (n)}_L$ (b) of type (A) (red) and of type (B) (dashed-blue).
• Figure 5: (a) Numerical "level crossing" of two eigenvalues of $K^{ (0)}_L$ for $L=4$ in the even twist parity sector. (b) The functions $D_{1,2}(p^2)$ defined in Eq. (\ref{['derivative']}).