Schnabl's Solution and Boundary States in Open String Field Theory
aa r X i v : . [ h e p - t h ] O c t RIKEN-TH-128UT-08-12April, 2008
Schnabl’s Solution and Boundary Statesin Open String Field Theory
Teruhiko Kawano ⋆ , Isao Kishimoto † and Tomohiko Takahashi ∗ ⋆ Department of Physics, University of Tokyo, Hongo, Tokyo 113-0033, Japan † Theoretical Physics Laboratory, RIKEN, Wako 351-0198, Japan ∗ Department of Physics, Nara Women’s University, Nara 630-8506, Japan
Abstract
We discuss that Schnabl’s solution is an off-shell extension of the boundary state describinga D-brane in the closed string sector. It gives the physical meaning of the gauge invariantoverlaps for the solution in our previous paper and supports Ellwood’s recent proposal in theoperator formalism.
In our previous paper [1], it was discussed that a class of gauge invariant observablesgives the same values for the analytic solution [2] given by Schnabl as the ones for thenumerical solution [3, 4, 5] in the level truncation in open string field theory [6]. Itgives another interesting evidence that the numerical solution is gauge equivalent tothe analytic solution.The gauge invariant observables O V (Ψ) for an open string field Ψ are called gaugeinvariant overlaps in [1] and are given by O V (Ψ) = h V ( i ) f I [Ψ] i = hI| V ( i ) | Ψ i , f I ( z ) ≡ z − z , where the CFT correlator is defined on an upper half plane. The on-shell closed stringvertex operator V ( i ) is inserted at the midpoint of the open string, and the conformalmapping f I ( z ) plays the role of the identity state hI| , identifying the left half of thestring with its right half. They were originally discussed in [7] to give the interactionof an on-shell closed string with open strings in the open string field theory.For the analytic solution | Ψ λ =1 i = | ψ i + ∞ X n =0 ( | ψ n +1 i − | ψ n i − ∂ r | ψ r i | r = n ) ≡ | ψ i + | χ i , O V ( ψ r ) doesn’t depend on r , one can see that O V (Ψ λ =1 ) = hI| V ( i ) | Ψ λ =1 i = hI| V ( i ) | ψ i . Therefore, it suggests that only the first term | ψ i contributes to the gauge invariantoverlaps O V (Ψ λ =1 ).In the paper [1], the gauge invariant overlaps were used to examine whether the nu-merical solution is gauge equivalent to the analytic solution, but their physical meaningwasn’t clear. Recently, Ellwood has made an interesting proposal [8] that the gaugeinvariant overlaps are related to the closed string tadpoles as O V (Ψ) = A Ψ ( V ) − A ( V ) , where A Ψ ( V ) is the disk amplitude for a closed string vertex operator V with theboundary condition of the CFT given by the open string field solution Ψ, and A ( V )is the usual disk amplitude in the perturbative vacuum.In fact, for the analytic solution, he has shown that this is the case. Since, aftertachyon condensation, there are no D-branes, no closed tadpoles are available, and thus A Ψ ( V ) = 0. Therefore, the gauge invariant overlap O V (Ψ) for the solution gives theusual disk amplitude of the opposite sign with one closed string emitted. It suggeststhat the analytic solution given by Schnabl is closely related to the boundary statedescribing a D-brane in the closed string perturbation theory, as expected.In this note, we will explicitly demonstrate this in the operator formalism of openstring field theory by using the Shapiro-Thorn vertex h ˆ γ (1 c , | [9], which maps an openstring state to the corresponding closed string state. (For more detail, see appendix Bin [1].) By using the open-closed string vertex h ˆ γ (1 c , | , as discussed in detail in [1],one may rewrite a gauge invariant overlap as O V ( ψ ) = h ˆ γ (1 c , | φ c i c | ψ i , where | φ c i c is an on-shell state given by the vertex operator V of the closed string 1 c ,and | ψ i is a state of the open string 2.For the analytic solution, one has O V (Ψ λ =1 ) = h ˆ γ (1 c , | φ c i c | Ψ λ =1 i = h ˆ γ (1 c , | φ c i c | ψ i + h ˆ γ (1 c , | φ c i c | χ i , and as mentioned above, the second term on the right hand side is zero for the on-shellclosed string state | φ c i c . As for the first term on the right hand side, one can see thatthe open string tachyon state | ψ i = (2 /π ) c | i is transformed via the vertex h ˆ γ (1 c , | into the boundary state h B | . In fact, one can obtain the relation h ˆ γ (1 c , | ψ i P c = 12 π h B | c − P c for the closed string 1 c , where the boundarystate is the usual one h B | = h | c − ¯ c − c +0 e − P ∞ n =1 [ n α n · ¯ α n + c n ¯ b n +¯ c n b n ] in the closed string perturbation theory. One thus finds that O V (Ψ λ =1 ) = 12 π h B | c − | φ c i , which is in precise agreement with Ellwood’s result.Since the second term h ˆ γ (1 c , | φ c i c | χ i doesn’t necessarily vanish for off-shellclosed string states, one may conclude that the transform of the analytic solution | Ψ λ =1 i via the Shapiro-Thorn vertex is an off-shell extension of the boundary state | B i .Although it seems more elaborate to calculate h ˆ γ (1 c , | χ i , it would be interesting tofind the relation of the off-shell boundary state with the equation of motion in closedstring field theory [10, 11, 12].Furthermore, given all the interactions between open strings and closed strings inthe open-closed string field theory [13], one may raise a question whether the Schnablsolution is consistent with the equations of motion of the open-closed string field theory,even in the vanishing string coupling constant limit, but it is beyond the scope of thispaper. However, if it is consistent, the relation of the transform of the analytic solution | Ψ λ =1 i via the Shapiro-Thorn vertex with the boundary state | B i could be clearer alongwith the interactions of the theory [13]. We think that our observation in this papermay serve as an encouraging step in the investigation. Acknowledgement
We are grateful to Taichiro Kugo for relevantly useful discussions. The work of T. K.was supported in part by a Grant-in-Aid ( More recently, the idea of this paper is extended to the marginal solutions [14] and the rollingtachyon solution [15] by one of the authors [16]. It may suggest that our observation of this papercould be more generic. eferences [1] T. Kawano, I. Kishimoto and T. Takahashi, “Gauge Invariant Overlaps for Clas-sical Solutions in Open String Field Theory,” Nucl. Phys. B , 135 (2008), arXiv:0804.1541 .[2] M. Schnabl, “Analytic Solution for Tachyon Condensation in Open String FieldTheory,” Adv. Theor. Math. Phys. , 433 (2006), hep-th/0511286 .[3] A. Sen and B. Zwiebach, “Tachyon Condensation in String Field Theory,” JHEP , 002 (2000), hep-th/9912249 .[4] N. Moeller and W. Taylor, “Level Truncation and the Tachyon in Open BosonicString Field Theory,” Nucl. Phys. B , 105 (2000), hep-th/0002237 .[5] D. Gaiotto and L. Rastelli, “Experimental String Field Theory,” JHEP , 048(2003), hep-th/0211012 .[6] E. Witten, “Noncommutative Geometry and String Field Theory,” Nucl. Phys. B , 253 (1986).[7] B. Zwiebach, “Interpolating String Field Theories,” Mod. Phys. Lett. A , 1079(1992), hep-th/9202015 .[8] I. Ellwood, “The Closed String Tadpole in Open String Field Theory,” JHEP , 063 (2008), arXiv:0804.1131 .[9] J. A. Shapiro and C. B. Thorn, “Closed String - Open String Transitions AndWitten’s String Field Theory,” Phys. Lett. B , 43 (1987).[10] M. Saadi and B. Zwiebach, “Closed String Field Theory from Polyhedra,” AnnalsPhys. , 213 (1989).[11] T. Kugo, H. Kunitomo and K. Suehiro, “Nonpolynomial Closed String Field The-ory,” Phys. Lett. B , 48 (1989);T. Kugo and K. Suehiro, “Nonpolynomial Closed String Field Theory: Action andits Gauge Invariance,” Nucl. Phys. B , 434 (1990).[12] B. Zwiebach, “Closed String Field Theory: Quantum Action and the B-V MasterEquation,” Nucl. Phys. B , 33 (1993), hep-th/9206084 .[13] B. Zwiebach, “Oriented Open-Closed String Theory Revisited,” Annals Phys. ,193 (1998), arXiv:hep-th/9705241 . 414] M. Schnabl, “Comments on marginal deformations in open string field theory,”Phys. Lett. B , 194 (2007) [arXiv:hep-th/0701248];M. Kiermaier, Y. Okawa, L. Rastelli and B. Zwiebach, “Analytic solutions formarginal deformations in open string field theory,” JHEP , 028 (2008)[arXiv:hep-th/0701249].[15] S. Hellerman and M. Schnabl, “Light-like tachyon condensation in Open StringField Theory,” arXiv:0803.1184 .[16] I. Kishimoto, “Comments on Gauge Invariant Overlaps for Marginal Solutions inOpen String Field Theory,” arXiv:0808.0355arXiv:0808.0355