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...nitiated their study in closed string field theory. In the course of this work we have learned that similar results to ours have been obtained independently by Kiermaier, Okawa, Rastelli and Zwiebach =-=[24]-=- and should appear in preprint at the same time as our work. 1 For related recent development see [5, 6, 7, 8, 9, 10]. Nice reviews of string field theory include [11, 12]. 22 Marginal deformations i...

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...question in a concrete setting. Analytic solutions for marginal deformations when operator products of the marginal operator are regular were constructed to all orders in the deformation parameter in =-=[17, 18]-=- for open bosonic string field theory [27] and in [19, 20, 22] for open superstring field theory [28]. When the operator product of the marginal operator is singular, analytic solutions were construct...

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...ircumference n is defined by first rescaling ˜z → 2 ˜z to get back n to a cylinder of width 2 and then mapping ˜z → f −1 (˜z) to get back to the upper half plane. We will often follow the notation of =-=[14]-=- and consider the fundamental region of the cylinder to be the region −1 1 < ℜ(˜z) < n − . This unusual choice happens to be convenient for the 2 2 form of some string field solutions. A prototypical ...

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... review recent remarkable progress in analytic methods for open string field theory [8]–[31], focusing on marginal deformations. Analytic solutions for marginal deformations were first constructed in =-=[20, 21]-=- for the bosonic string when operator products of the marginal operator are regular, and the solutions were extended to the superstring in [22, 23, 25]. The generalization to marginal deformations wit...

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...ions to string field theory was performed in [186]. Following this work other papers appeared [187,188,32,189,190,191,192,193,194]. The first analytical solutions of marginal deformations appeared in =-=[195,196]-=-, where the solution describing general marginal deformations with regular OPE’s was given. The OPE whose regularity we refer to is that of V with itself. Thus, the simple analytical solutions given c...

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... by using the techniques employed in Schnabl’s solution (in particular, the B0 gauge), yielding exactly marginal deformations order by order in a parameter λ parameterizing the exactly flat direction =-=[30, 31]-=-. This approach was generalized [32, 33] to describe also the first analytical solutions of superstring field theory [34]. – 1 – J H E P09(2007)101 The approach of [30, 31] gives an explicit solution ...

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...rom numerical studies [6–13]. Adding to this OSFT revival is the construction of an exact solution representing the dynamical rolling of the tachyon from the perturbative vacuum to the tachyon vacuum =-=[14,15]-=-. Although we will focus on the bosonic case, a rolling-tachyon solution has also been constructed for Berkovits’ supersymmetric open string field theory [16–18] in [19–21]. See also [22] for another ...

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... section, the tachyon vacuum is given by the state Φ = 0 and hence ΩΦ = PΦ = 0. 4.2 Marginal deformations with trivial OPE For marginal deformations with trivial OPE, a B0-gauge solution was found in =-=[8,9]-=-. It is given by 1 Θ = −Ψ + Θ0 = −Ψ − W1/2 ∗ 1 − λ cJ ∗ A ∗ λ cJ ∗ W1/2 . (4.4) Here we are using the following notation: Wr is the wedge state [27] of width r (with r = 1 the SL2(R) vacuum) . As befo...

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... B L 1 φ1 ∗ φ2 = B1φ1 ∗ φ2 + (−1) gh(φ1) φ1 ∗ B L 1 φ2, {B1, c1} = 1, B1|0〉 = 0, (B L 1 )2 = 0. Recently, marginally deformed exact solutions around the unstable vacuum are obtained in bosonic string =-=[14, 15, 18, 19, 22, 25]-=- and superstring [16, 17, 22, 23, 28] field theories. If the operator V has a regular OPE with itself, the solutions are well-defined. However, when the OPE of V is singular, divergencies aries in the...