Comments on Schnabl's analytic solution for tachyon condensation in Witten's open string field theory

Abstract

Schnabl recently constructed an analytic solution for tachyon condensation in Witten's open string field theory. The solution consists of two pieces. Only the first piece is involved in proving that the solution satisfies the equation of motion when contracted with any state in the Fock space. On the other hand, both pieces contribute in evaluating the kinetic term to reproduce the value predicted by Sen's conjecture. We therefore need to understand why the second piece is necessary. We evaluate the cubic term of the string field theory action for Schnabl's solution and use it to show that the second piece is necessary for the equation of motion contracted with the solution itself to be satisfied. We also present the solution in various forms including a pure-gauge configuration and provide simpler proofs that it satisfies the equation of motion.

Tachyon solution in a cubic Neveu-Schwarz string field theory

“…This proof is analogous to calculations performed by Okawa [19] for the bosonic string and is based on equations from Appendix.…”

“…As it now well understood [19,22,27] to calculate the action on a solution one has to provide the validity of the equation of motion in a strong sense. In other words one has to regulates the constructed formal solution.…”

“…Finally, Schnabl found an analytical solution in the open bosonic Witten's SFT and proved analytically the tachyon condensation [18]. His solution and related problems have been examined in a series of recent works and its algebraic structure has been studied in detail [19,20,21,22,23,24,25]. Construction of lower D-branes solutions and the analytical proof of the relations between D-brane tensions are also a subject of a recent interest.…”

Analytic solutions for marginal deformations in open superstring field theory

“…We follow the notation used in [3,10,17]. In particular, see the beginning of section 2 of [3] for the relation to the notation used in [2].…”

Schnabl's Script L0operator in the continuous basis

“…Yet, for this calculation, the dependence on the matter sector simply amounts to integrating over the zero-modes, which gives the needed 26 dimensional volume factor. Therefore, Schnabl was able to prove Sen's first conjecture within the universal basis (up to some subtleties that were clarified in [4,5]). …”