Test of the Absence of Kinetic Terms around the Tachyon Vacuum in Cubic String Field Theory

TL;DR

The paper investigates whether the cubic open string field theory around the non-perturbative tachyon vacuum lacks open string dynamics by examining the absence of kinetic terms for BRST-invariant fluctuations within a level truncated, derivative-expanded framework. It derives the derivative expansion of the action and BRST transformation in a level (2,6) truncation restricted to scalar fluctuations, and then numerically analyzes the two- and four-derivative kinetic terms, along with BRST consistency, near the vacuum. The results provide partial evidence that the two-derivative kinetic term can vanish for the BRST-invariant combination, with zero-modes of the kinetic matrices approaching the non-perturbative vacuum, though higher-derivative terms and full BRST structure do not conclusively cancel in this truncation. The work suggests the conjecture of no open string excitations at the tachyon vacuum is plausible within a more complete truncation but highlights substantial limitations and identifies clear directions for future refinement, including incorporating vector and tensor modes and higher truncations. Overall, the study contributes to understanding how open string dynamics may disappear at the tachyon vacuum and hints at a possible BRST/topological character of the shifted theory, with implications for the relation between open and closed string sectors.

Abstract

It has been conjectured that the bosonic open string theory around the non-perturbative tachyon vacuum has no open string dynamics at all. We explore, in the cubic open string field theory with level truncation approximation, the possibility that this conjecture is realized by the absence of kinetic terms of the string field fluctuations. We study the kinetic terms with two and four derivatives for the lower level scalar modes as well as their BRST transformation properties. The behavior of the coefficients of the kinetic terms in the neighborhood of the non-perturbative vacuum supports our expectation that the BRST invariant scalar component lacks its kinetic term.

Figures (7)

• Figure 1: $\det G$ as a function of $t$ for $(u,v)=(0,0),(0.1,0.1),(0.2,0.2),(0.3,0.3)$.
• Figure 2: Curves of $\det G=0$ in the $(u,v)$ plane for $t=0.4,0.45,0.5,0.55,0.6$.
• Figure 3: $\det H$ as a function of $t$ for $(u,v)=(0,0),(u_r,v_r),(0.15,0.30),(0.15,0.35)$.
• Figure 4: Curves of $\det H=0$ in the $(v,u)$ plane for $t=0.4,0.5,0.6$.
• Figure 5: $|\bm{f}_1\cdot\bm{e}_0^r|$ and $|\bm{f}_2\cdot\bm{e}_0^r|$ as a function of $v$ on the surface $\det H=0$ and with $t=t_r$.