Observables for identity-based tachyon vacuum solutions

Abstract

We consider a modified KBc algebra in bosonic open string field theory expanded around identity-based scalar solutions. By use of the algebra, classical solutions on the background are constructed and observables for them, including energy densities and gauge invariant overlaps, are calculable. These results are applied to evaluate observables analytically for both of the identity-based trivial pure gauge solution and the identity-based tachyon vacuum solution. §1. IntroductionAn analytic tachyon vacuum solution was constructed on the basis of the identity string field, the BRST current, and the ghost field in bosonic cubic open string field theory. 1), 2), 3) The identity string field is a fundamental object in the open string field theory 4) and indeed it is a building block of the KBc algebra 5) by which wedge-based solutions 6) can be easily reconstructed. Then, the identity-based solutions were found by some left-right splitting algebra, 1) which is similar to the KBc algebra in a sense, and a certain type of the identitybased solutions can be regarded as the tachyon vacuum solution. This is supported by evidence from study of the theory expanded around the solution: vanishing cohomology, 2), 3) , 7) no open string excitations, 8) and the existence of the perturbative vacuum solution. 9), 10) Hence, it appears highly probable that observables for the identity-based tachyon vacuum solution agree with those expected for the tachyon vacuum, although, due to characteristic subtleties of the identity string field, it has been difficult to perform direct evaluation of the observables.Recently, significant progress has been made in the investigation of identity-based marginal solutions. We have obtained a gauge equivalence relation including the identity-based marginal solutions and some kind of wedge-based tachyon vacuum solutions and, using this relation, we can directly evaluate observables for the identify-based solutions. 11), 12) The key ingredient is a combined technique for the identity-based solutions and the KBc algebra 13), 14) and it has potentiality for investigating string field theory. In fact, it has been applied to construct a new solution, 15) which has the same algebraic structure as a wedge-based marginal solution 16) and is gauge equivalent to the identity-based marginal solution.The main purpose of this paper is, based on these developments, to confirm directly that the identity-based scalar solution provides the correct observables as expected.The identity-based scalar solution is given by 1)where Q L (f ) and C L (f ) are integrations of the BRST current j B (z) and the ghost c(z), which are multiplied by a function f (z) along a half unit circle. We find that the equation of motion holds for the function h(z) such that h(−1/z) = h(z) and h(±i) = 0. Moreover, the reality condition of (1 . 1) imposes the function h(z) to satisfy (h(z)) * = h(1/z * ).Expanding the string field Ψ around the solution as Ψ = Ψ 0 + Φ, we obtain an action for fluctuation:

Closed string symmetries in open string field theory: tachyon vacuum as sine-square deformation

“…As a result, the theory expanded around the solution includes the modified BRST operator expressed as the integration of local operators on a worldsheet, which is a useful tool for investigating the corresponding vacuum. Actually, we can find the perturbative vacuum in the theory expanded around the identity-based tachyon vacuum solution with high precision by a level truncation scheme [16,17,18], although the vacuum energy is analytically calculated later [19,20,21]. This is a remarkable feature that has never been seen in wedge-based solutions [22,23].…”

Comments on Takahashi-Tanimoto’s scalar solution

“…In the workshop "String field theory and related aspects VI, SFT2014" (July 28 -August 1, 2014, SISSA Italy), where this work is presented [20], we have learned that Kishimoto, Masuda and Takahashi work on the same problem from a different point of view [21] [22]. Their results have some overlap with those in section 2.…”

“… Kishimoto, Masuda and Takahashi [22] generalize the method of[28,32] to the case of the scalar solutions 3. Kishimoto, Masuda and Takahashi[22] obtain the same results using a different method, considering more general solutions made from K ′ Bc.…”

A singular one-parameter family of solutions in cubic superstring field theory

“…It is known that the analytic solutions for tachyon condensation [1,2,3] in open bosonic string field theory [4] as well as the ones [5,6,7] in cubic superstring field theory [8] are formally gauge equivalent to identity based solutions [9,10,11,12,13]. Identity based solutions are constructed as a product of certain linear combination of ghost number one operators with the identity string field [14,15,16].…”

“…Since a string field belonging to the GSO(−) sector has half-integer conformal weight, Φ 2 changes its sign under the conformal transformation R 2π representing the 2π rotation of the unit disk[28] 3. Although the identity based solution formally satisfies the equation of motion (2.9), it is a pathological solution in the sense that it brings ambiguous analytic result for the value of the energy[9,14,15,16].…”