Analytic study of nonperturbative solutions in open string field theory

Abstract

We propose an analytic framework to study the nonperturbative solutions of Witten's open string field theory. The method is based on the Moyal star formulation where the kinetic term can be split into two parts. The first one describes the spectrum of two identical half strings which are independent from each other. The second one, which we call midpoint correction, shifts the half string spectrum to that of the standard open string. We show that the nonlinear equation of motion of string field theory is exactly solvable at zeroth order in the midpoint correction. An infinite number of solutions are classified in terms of projection operators. Among them, there exists only one stable solution which is identical to the standard butterfly state. We include the effect of the midpoint correction around each exact zeroth order solution as a perturbation expansion which can be formally summed to the complete exact solution.Recently, this formal correspondence played a major rôle in importing basic ideas of noncommutative geometry to string field theory. One of the stimulating ideas is the vacuum string field theory (VSFT) proposal [3]. With an assumption on a simplified kinetic term to describe the tachyon vacuum, the classical solutions that would describe the D-brane are given by the noncommutative soliton [4] (=projector). It is well known that projectors are the fundamental geometrical objects in noncommutative geometry since they represent the K-homology group. The proof of the VSFT conjecture on the kinetic term was, however, difficult and there remained many open questions.The kinetic term is given by the Virasoro operator which is a second order differential operator acting on string fields A x, ξ, ξ gh . We specify this operator later since it will be

Solving Witten’s string field theory using the butterfly state

“…We use the butterfly state in a different way. Our expansion scheme and lowest-order solution are different from those of [32].…”

“…2 See Appendix A of [26] for a check of this relation following Polchinski's conventions in [27]. 3 An analytic framework to solve the equation (L 0 − 1) |Ψ + |Ψ ⋆ Ψ = 0 using the butterfly state was proposed in [32]. Here ⋆ is the Moyal star product [33,34,35,36,37,32,38], and a closed expression of a formal exact solution has been provided as a perturbative expansion.…”

“…3 An analytic framework to solve the equation (L 0 − 1) |Ψ + |Ψ ⋆ Ψ = 0 using the butterfly state was proposed in [32]. Here ⋆ is the Moyal star product [33,34,35,36,37,32,38], and a closed expression of a formal exact solution has been provided as a perturbative expansion. We use the butterfly state in a different way.…”

Exact Solutions in SFT and Marginal Deformation in BCFT

“…Much work has focused on the search for classical solutions of cubic bosonic open string field theory (SFT) [1] (For review, see [2,3,4,5].). Despite important technical progress in the understanding of the open string star product-notably the discovery of a new connections with non-commutative field theory [6,7,8,9,10,11,12,13,14,15] it is still very difficult to find analytic classical solution. Exception is the special form of SFT, vacuum string field theory (VSFT), where exact results have been obtained [18,19,20,21,22,23,24,25,26].…”

Vacuum string field theory without matter-ghost factorization

“…See also[27] 6. A different analytic framework to solve Witten's string field theory using the butterfly state was proposed earlier in[28].…”