Numerical solution of open string field theory in Schnabl gauge

Abstract

Using traditional Virasoro L 0 level-truncation computations, we evaluate the open bosonic string field theory action up to level (10,30). Extremizing this level-truncated potential, we construct a numerical solution for tachyon condensation in Schnabl gauge. We find that the energy associated to the numerical solution overshoots the expected value −1 at level L = 6. Extrapolating the level-truncation data for L ≤ 10 to estimate the vacuum energies for L > 10, we predict that the energy reaches a minimum value at L ∼ 12, and then turns back to approach −1 asymptotically as L → ∞. Furthermore, we analyze the tachyon vacuum expectation value (vev), for which by extrapolating its corresponding level-truncation data, we predict that the tachyon vev reaches a minimum value at L ∼ 26, and then turns back to approach the expected analytical result as L → ∞.

Numerical solution for tachyon vacuum in the Schnabl gauge

“…In this work, we have managed to solve the aforementioned technical issues, and we have obtained results up to level L = 24, using a clever numerical method based on the traditional level truncation scheme, which in principle can be applied to all general linear b-gauges. We have explicitly proven that the energy of the numerical solution has in fact a local minimum at level L = 12, so the conjecture made in [36] was proven to be correct.…”

“…One of the main motivations of this work is to provide a conclusive evidence of the conjecture in reference [2], that the numerical solution constructed in the Schnabl gauge by means of level truncation computations can be identified with the analytical solution (1.1) when L → ∞. An obvious step to accomplish this task is to perform higher level computations, this might appear as an straightforward extension of the calculations developed in reference [36].…”

“…The equations (1.1) and (1.2) can be used to expand the analytical solution in the state space of Virasoro L 0 JHEP02(2020)065 10 0.546508411314 −1.008189759705 Table 1. (L, 3L) level truncation results of reference [36] for the tachyon vev and vacuum energy in the Schnabl gauge up to level L = 10.…”

“…In reference [2], the author conjectured that the level dependent Schnabl gauge fixing condition would not pose problems and using the high L 0 level truncation computations of Moeller and Taylor [34] and Gaiotto and Rastelli [35], it should be possible to construct a numerical solution that would converge to his analytical solution when the level goes to infinity. The first attempt to obtain a numerical solution for tachyon vacuum in the Schnabl gauge was made by Arroyo et al [36], using the traditional level truncation computations up to level L = 10. By extrapolating the energy data of levels L ≤ 10, shown in table 1, to estimate the energy for L > 10, the authors predicted that the energy reaches a local minimum value at level L = 12, to subsequently turn back to approach −1 asymptotically as L → ∞.…”

“…However the numerical method used in reference [36] is not practical for levels beyond L > 12. To see why, let us briefly explain how it works.…”