Observables for identity-based tachyon vacuum solutions
aa r X i v : . [ h e p - t h ] D ec Observables for identity-based tachyon vacuumsolutions
Isao
Kishimoto , Toru Masuda and Tomohiko Takahashi Faculty of Education, Niigata University, Niigata 950-2181, Japan Department of Physics, Nara Women’s University, Nara 630-8506, Japan
Abstract
We consider a modified
KBc algebra in bosonic open string field theory expandedaround identity-based scalar solutions. By use of the algebra, classical solutions onthe background are constructed and observables for them, including energy densitiesand gauge invariant overlaps, are calculable. These results are applied to evaluateobservables analytically for both of the identity-based trivial pure gauge solution andthe identity-based tachyon vacuum solution.1
1. Introduction
An analytic tachyon vacuum solution was constructed on the basis of the identity stringfield, the BRST current, and the ghost field in bosonic cubic open string field theory.
The identity string field is a fundamental object in the open string field theory and indeedit is a building block of the KBc algebra by which wedge-based solutions can be easilyreconstructed. Then, the identity-based solutions were found by some left-right splittingalgebra, which is similar to the KBc algebra in a sense, and a certain type of the identity-based solutions can be regarded as the tachyon vacuum solution. This is supported by evi-dence from study of the theory expanded around the solution: vanishing cohomology, no open string excitations, and the existence of the perturbative vacuum solution. Hence, it appears highly probable that observables for the identity-based tachyon vacuumsolution agree with those expected for the tachyon vacuum, although, due to characteristicsubtleties of the identity string field, it has been difficult to perform direct evaluation of theobservables.Recently, significant progress has been made in the investigation of identity-based marginalsolutions. We have obtained a gauge equivalence relation including the identity-basedmarginal solutions and some kind of wedge-based tachyon vacuum solutions and, using thisrelation, we can directly evaluate observables for the identify-based solutions.
The keyingredient is a combined technique for the identity-based solutions and the
KBc algebra and it has potentiality for investigating string field theory. In fact, it has been applied to con-struct a new solution, which has the same algebraic structure as a wedge-based marginalsolution and is gauge equivalent to the identity-based marginal solution.The main purpose of this paper is, based on these developments, to confirm directly thatthe identity-based scalar solution provides the correct observables as expected.The identity-based scalar solution is given by Ψ = Q L ( e h − I − C L (( ∂h ) e h ) I, (1.1)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 equationof motion holds for the function h ( z ) such that h ( − /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 + Φ , we obtain an action forfluctuation: S [ Ψ ; Q B ] = S [ Ψ ; Q B ] + S [ Φ ; Q ′ ] , (1.2)2here we denote the action as S [ Ψ ; Q ] = − R (cid:0) Ψ ∗ QΨ + Ψ ∗ Ψ ∗ Ψ (cid:1) and the kinetic oper-ator Q ′ is given by Q ′ = Q ( e h ) − C (( ∂h ) e h ) . (1.3)The operators Q ( f ) and C ( f ) are defined as integrations along a whole unit circle.We have a degree of freedom to choose a function h ( z ) in the classical solution and itcan be changed by gauge transformations. Since the function continuously connects to zero,most of the solutions are regarded as a trivial pure gauge solution. However, nontrivialsolutions are generated at the boundary of some function spaces. In the well studied case,the function includes one parameter a ≥ − / h a ( z ) = log (cid:16) a z + z − ) (cid:17) . (1.4)It is known that the solution for a > − / a = − /
2, for the reason mentioned above.The transition from a trivial pure gauge to the tachyon vacuum solution has been observedin various aspects of the identity-based solution. In consequence, it is known that zeros of e h ( z ) move on the z plane with the deformation of h ( z ) and then the transition occurs whenthe zeros reach the unit circle | z | = 1. For example, e h ( z ) for (1.4) is rewritten as e h a ( z ) = 1(1 − Z ( a )) { z + Z ( a ) }{ z − + Z ( a ) } (cid:18) Z ( a ) = 1 + a − √ aa (cid:19) , (1.5)and it has zeros at ± p − Z ( a ) and ± / p − Z ( a ). When the parameter a approaches − / Z ( a ) runs from 1 to − − a = − / a = − / For other functions, we find that the sametransition occurs if the zeros move to the unit circle. We now briefly outline our strategy. First, we find the
KBc algebra in the shiftedtheory with the action S [ Φ ; Q ′ ], which we call the K ′ Bc algebra. By means of the K ′ Bc algebra, it is straightforward to construct classical solutions in the shifted theory and it ispossible to calculate observables for these solutions. Here, the shifted theory includes oneparameter a , as the above example, through Q ′ , and so the classical solutions depend onthe parameter. This is similar to the case of the analysis for the identity-based marginalsolutions, in which the shifted theory and the solution include parameters related tomarginal deformations. Therefore, according to the marginal case, we represent the identity-based solution as a gauge equivalence relation involving the identity-based and wedge-based3olutions. Finally, by use of this expression, we evaluate observables directly for the identity-based solution.Later we will see that there is a difference between the K ′ Bc algebra around the identity-based trivial solution and that around the identity-based nontrivial solution. If Ψ is a trivialpure gauge solution, the K ′ Bc algebra can be transformed to the original KBc algebra.With the help of the transformation, we can calculate observables for classical solutionsin the shifted theory. Here, the existence of such a transformation depends crucially onthe positions of zeros of the function e h ( z ) . The zeros on the unit circle become obstacles toconstruction of the transformation and therefore it is impossible to transform from the K ′ Bc algebra to the KBc one in the case that Ψ is the tachyon vacuum solution. However, thisimplies that, on the identity-based tachyon vacuum, K ′ , B , and c have a different algebraicstructure from the original KBc algebra. We will find that, on the identity-based tachyonvacuum, the operators K ′ and c commute with each other and then all the solutions madeof K ′ , B and c can be written as modified BRST exact states. Accordingly, observables forthem are calculable even if Ψ is the tachyon vacuum solution.This paper is organized as follows: First, we will consider classical solutions in the theoryexpanded around the identity-based solution in Sect. 2. We construct the K ′ Bc algebrawith respect to Q ′ and, by using the K ′ Bc algebra, we will find classical solutions on theidentity-based vacuum. To calculate observables for the classical solutions, we will constructa similarity transformation from the operator ( K ′ ) L to ( K ) L . We will find that a conformaltransformation is a significant part of the similarity transformation and so we will illustrateit by an example for h a ( z ) in (1.4). Then, we will calculate observables for the classicalsolutions around Ψ . In Sect. 3, based on the results in the previous section, we analyticallyevaluate observables for the identity-based solutions. In Sect. 4, we will give concludingremarks. In Appendix A, we provide a detailed proof of the properties of a differential equa-tion that plays an important role in the calculation of observables. Note added:
When we had a discussion with N. Ishibashi during the conference SFT2014 atSISSA, Trieste, it was found that we had reached the same conclusion for the gauge invariantobservables for the identity-based tachyon vacuum solution. ∗ ) The main difference is that heargued in detail for a regularization method to evaluate the observables but we evaluatedthe observables for the identity-based trivial solution in addition to the tachyon vacuumcase. ∗ ) Both Ishibashi’s and our results were presented independently at the conference. The presentationfiles of these talks by N. Ishibashi and one of the authors (T.T.) are available on the conference website: . which treatssimilar problems with different methods. §
2. Classical solutions around the identity-based solution
Modified
KBc algebra
We can construct a modified
KBc algebra associated with the deformed BRST operator(1.3): K ′ = Q ′ B, Q ′ K ′ = 0 , Q ′ c = cK ′ c, (2.1) B = 0 , c = 0 , Bc + cB = 1 , (2.2)where B and c are the same string fields in the conventional KBc algebra, and K ′ is givenby ∗∗ ) K ′ = π K ′ ) L I, ( K ′ ) L = { Q ′ , ( B ) L } . (2.4)The operator ( K ′ ) L is explicitly calculated as( K ′ ) L = Z C left dz πi (1 + z ) (cid:26) e h ( z ) T ( z ) + ( ∂h ) e h ( z ) j gh ( z ) + (cid:18) ∂ h + 12 ( ∂h ) (cid:19) e h ( z ) (cid:27) , (2.5)where j gh ( z ) is the ghost number current and T ( z ) is the total energy momentum tensor. ∗ ) We can easily find that if h ( z ) becomes identically zero, the operator ( K ′ ) L is equal to theconventional ( K ) L in the KBc algebra.For the general function h ( z ), K ′ , B , c , and Q ′ have the same algebraic structure as thatof the KBc algebra. However, if we choose a special function, the algebra is more simplified.To see this let us consider the relation Q ′ c = cK ′ c in (2.1). This relation is derived from thefollowing equations: { Q ( e h ) , c ( z ) } = e h ( z ) c∂c ( z ) , (2.6)[ K ′ , c ( z )] = − ( ∂ (1 + z )) e h ( z ) c ( z ) + (1 + z ) e h ( z ) ∂c ( z ) , (2.7) ∗∗ ) We use the following convention: B = π B ) L I, c = 1 π c (1) I, ( B ) L = Z C left dz πi (1 + z ) b ( z ) , (2.3)where I is the identity string field and the integration path C left on the z -plane is a half unit circle: | z | =1 , Re z ≥ ∗ ) It can be calculated by using the relations in Ref. 7): { Q ( f ) , b ( z ) } = 32 ∂ f ( z ) + ∂f ( z ) j gh ( z ) + f ( z ) T ( z ) , { C ( f ) , b ( z ) } = f ( z ) . K ′ is the operator defined by the replacement of the integration path in (2.5) with aunit circle: K ′ = { Q ′ , B } = { Q ′ , b + b − } . As mentioned in Sect. 1, the function e h ( z ) haszeros on the unit circle in the case that the solution becomes the tachyon vacuum solution.Indeed, for (1.4), e h a ( z ) has zeros at z = ± a = − / Q ( e h a ) and c (1) anticommute with each other for a = − / ∗ ) Similarly, K ′ and c (1)commute for a = − / e h ( z ) has zeros at z = ±
1, namely in the theory around the tachyon vacuum solution: K ′ = Q ′ B, Q ′ K ′ = 0 , Q ′ c = 0 , (2.8) B = 0 , c = 0 , Bc + cB = 1 . (2.9)Actually, there are other possibilities where the function e h ( z ) for the tachyon vacuumsolution has zeros on the unit circle but not at z = ±
1. We will discuss these cases at theend of the section.2.2.
Classical solutions
The equation of motion in the theory around the solution (1.1) is given by Q ′ Φ + Φ = 0 , (2.10)where Q ′ is the modified BRST operator (1.3). We can find various classical solutions in theshifted background by substituting K ′ for K in the solutions given by the KBc algebra inthe original theory. In the conventional theory with Q B , a classical solution using the KBc algebra is written as Ψ ( K, B, c ) = X ij A i ( K ) c B j ( K ) + X ijk C i ( K ) c D j ( K ) c E k ( K ) B, (2.11)which is general configuration with ghost number one in terms of the KBc algebra. Here, A i ( K ) , B i ( K ) , C i ( K ) , D i ( K ), and E i ( K ) are appropriate functions of the string field K .Once a particular solution (2.11) is given, a classical solution for (2.10) is constructed as Φ ( K ′ , B, c ) = X ij A i ( K ′ ) c B j ( K ′ ) + X ijk C i ( K ′ ) c D j ( K ′ ) c E k ( K ′ ) B. (2.12)If Ψ ( K, B, c ) is a solution in the conventional theory, Φ ( K ′ , B, c ) is a solution in theshifted background, regardless of whether or not the K ′ Bc algebra is simplified as (2.8).However, we emphasize that in the case that the algebra is simplified, the solution has asimpler expression: Φ ( K ′ , c ) = F ( K ′ ) c, (2.13) ∗ ) We note that e − iσ c ( e iσ ) I = − e − i ( π − σ ) c ( e i ( π − σ ) ) I and therefore c (1) I = c ( − I . F ( K ′ ) = P ij A i ( K ′ ) B j ( K ′ ) and the second term in (2.12) vanishes due to K ′ c = cK ′ and c = 0.2.3. Transformations from ( K ′ ) L to ( K ) L In this subsection, we will consider a similarity transformation from ( K ′ ) L to the con-ventional ( K ) L .First, we introduce the operator ∗ ) ˜ q ( h ) = I dz πi h ( z ) (cid:18) j gh ( z ) − z − (cid:19) , (2.14)where j gh ( z ) is the ghost number current, j gh = cb . Using this operator, the modified BRSToperator (1.3) is transformed to the original BRST operator: ∗∗ ) e − ˜ q ( h ) Q ′ e ˜ q ( h ) = Q B . (2.15)Accordingly, we can remove the ghost number current from ( K ′ ) L in (2.5) by a similaritytransformation: e − ˜ q ( h ) ( K ′ ) L e ˜ q ( h ) = e − ˜ q ( h ) { Q ′ , ( B ) L } e ˜ q ( h ) = (cid:8) Q B , e − ˜ q ( h ) ( B ) L e ˜ q ( h ) (cid:9) = Z C left dz πi (1 + z ) e h ( z ) T ( z ) , (2.16)where we have used e − ˜ q ( h ) b ( z ) e ˜ q ( h ) = e h ( z ) b ( z ). Next, we look for a conformal transformation z ′ = f ( z ) that maps (2.16) to ( K ) L . Since T ( z ) is a primary field with the dimension 2, the operator (2.16) is transformed as f (cid:20)Z C left dz πi (1 + z ) e h ( z ) T ( z ) (cid:21) = Z C ′ left df πi (1 + z ) e h ( z ) dfdz T ( f ( z )) , (2.17)where C ′ left is an integration path in the mapped plane such as f : C left → C ′ left . In orderthat (2.17) coincides with ( K ) L , the function f ( z ) must satisfy a differential equation:(1 + z ) e h ( z ) dfdz = 1 + f , (2.18)and C ′ left must remain the same path along the left half of a string.To find the conformal map, it is necessary to solve the differential equation (2.18) in anannulus including the unit circle | z | = 1. The important point is that we can solve it if e h ( z ) ∗ ) This operator was written as K ( h ) in Ref. 8). The ghost number current j gh ( z ) is defined by using SL (2 , R ) normal ordering. If h ( z ) satisfies h ( − /z ) = h ( z ), owing to the second term ( − / z − ), theoperator is transformed as ˜ q ( h ) → − ˜ q ( h ) under the BPZ conjugation. Moreover, ˜ q ( h ) is a derivation withrespect to the star product among string fields. ∗∗ ) The operator e ± ˜ q ( h ) becomes singular for the tachyon vacuum solution. f (1) = 1, the solution f ( z ) has the following properties:1 . | z | = 1 ⇒ | f ( z ) | = 1 , (2.19)2 . f : C left ։ C left , (2.20)3 . f (cid:18) − z (cid:19) = − f ( z ) . (2.21)We illustrate these by the solution given for h a ( z ) (1.4) in the next subsection and we givea detailed proof in Appendix A.From (2.19) and (2.20), we find that the conformal map by the solution f leaves theintegration path in (2.17) unchanged, namely C ′ left = C left . Therefore we can transform theoperator (2.16) to ( K ) L by the conformal transformation f . Moreover, (2.21) indicates thatthe conformal map f ( z ) is generated by the operators K n = L n − ( − n L − n . ∗ ) Consequently, around the trivial pure gauge solution, we can construct the similaritytransformation U f e − ˜ q ( h ) ( K ′ ) L e ˜ q ( h ) U − f = ( K ) L , (2.22)where U f is the operator for the conformal transformation f and it is given in the form U f = exp X n v n K n ! , (2.23)with certain parameters v n .For the identity-based tachyon vacuum solution, a solution to the differential equation(2.18) for f ( z ) has singularity due to zeros of e h ( z ) on the unit circle (Appendix A). In thissense, we emphasize that a regular operator U f does not exist for the tachyon vacuum.2.4. An example for the transformation
We illustrate the existence of the transformation U f by solving (2.18) for (1.4). For (1.4), e h ( z ) is written as (1.5), and, under the initial condition f (1) = 1, setting z = e iσ , we cansolve the differential equation (2.18) as follows: f ( e iσ ) = e iφ ( σ ) , φ ( σ ) = σ + 2 arctan g ( σ ) cos σ g ( σ ) sin σ , (2.24)where, for − / < a ≤ − < Z ( a ) ≤ g ( σ ) is given as g ( σ ) = tanh ( p − Z ( a )1 + Z ( a ) arctan p − Z ( a )1 + Z ( a ) sin σ !) , (2.25) ∗ ) K n generates a transformation f ( σ ) such that f ( π − σ ) = π − f ( σ ). By setting z = e iσ , thiscorresponds to (2.21). < a (0 < Z ( a ) < g ( σ ) = − tanh ( p Z ( a )1 + Z ( a ) arctanh p Z ( a )1 + Z ( a ) sin σ !) . (2.26)Since φ ( σ ) is a real-valued function for σ ∈ R , the solution (2.24) satisfies (2.19). Bydifferentiating φ ( σ ), it can be seen that φ ( σ ) is a monotonically increasing function for − π/ < σ < π/
2. We also see that φ ( ± π/
2) = ± π/
2. Therefore, C ′ left is the same as theleft half of a string, then (2.20) is satisfied. Moreover, since g ( π − σ ) = g ( σ ), we find that φ ( π − σ ) = π − φ ( σ ) and then the function (2.24) satisfies (2.21). Thus, the solution (2.24)satisfies (2.19), (2.20) and (2.21) in the case of a > − /
2, and then the transformation U f exists.Here, we should emphasize that the transformation (2.23) exists only in the case a > − / a = − /
2, because the circle-to-circle correspondence for the integrationpath is broken down for a = − /
2. In fact, taking the limit a → − /
2, the phase φ ( σ ) in(2.24) approaches a step function:lim a →− / φ ( σ ) = π < σ < π ) − π − π < σ < . (2.27)Therefore, we cannot transform ( K ′ ) L to ( K ) L by a regular conformal map at a = − / Observables around the trivial pure gauge solution
In this subsection, we will show that observables for the solution (2.12) around theidentity-based trivial pure gauge solution are equivalent to those for the original solution(2.11).First, we find that c (1) and ( B ) L are invariant under the similarity transformation(2.22), namely, U f e − ˜ q ( h ) c (1) e ˜ q ( h ) U − f = c (1) and U f e − ˜ q ( h ) ( B ) L e ˜ q ( h ) U − f = ( B ) L . Using e − ˜ q ( h ) c ( z ) e ˜ q ( h ) = e − h ( z ) c ( z ), we have U f e − ˜ q ( h ) c ( z ) e ˜ q ( h ) U − f = e − h ( z ) (cid:18) df ( z ) dz (cid:19) − c ( f ( z )) . (2.28)From the differential equation (2.18), it follows that= 1 + z f ( z ) c ( f ( z )) , (2.29)and then c (1) is invariant under the transformation because f (1) = 1 is imposed as the initialcondition. With regard to ( B ) L , the invariance can be easily seen by using e − ˜ q ( h ) b ( z ) e ˜ q ( h ) = e h ( z ) b ( z ) and the fact that b ( z ) is a primary field with the dimension 2.9ow that the similarity transformation of ( K ′ ) L , ( B ) L , and c (1) is established, we cantransform the solution (2.12) to the original solution (2.11). An important point is that thegenerators ˜ q ( h ) and K n are derivations with respect to the star product and in particular˜ q ( h ) I = 0 and K n I = 0. Then, we obtain the transformation from string fields ( K, B, c ) to( K ′ , B, c ): K ′ = e ˜ q ( h ) U − f K, B = e ˜ q ( h ) U − f B, c = e ˜ q ( h ) U − f c. (2.30)Noting that U − f and e ˜ q ( h ) are given as an exponential of derivations, we find that the solution(2.12) is given as a transformation from (2.11): Φ ( K ′ , B, c ) = e ˜ q ( h ) U − f Ψ ( K, B, c ) . (2.31)Let us consider the vacuum energy for Φ around the trivial pure gauge solution. Usingthe transformation (2.31), the action for Φ is given by S [ Φ ( K ′ , B, c ); Q ′ ] = S [ Ψ ( K, B, c ); U f Q B U − f ] , (2.32)where we have used (2.15) and the BPZ conjugation: e ˜ q ( h ) U − f → U f e − ˜ q ( h ) . Since U f isgenerated by K n and the operators Q B and L n commute with each other, U f Q B U − f is equalto Q B . As a result, the vacuum energy for Φ ( K ′ , B, c ) is equivalent to that for Ψ ( K, B, c )in the conventional theory.Next, let us consider gauge invariant overlaps for Φ ( K ′ , B, c ). The gauge invariantoverlap for the open string field Ψ is defined as O V ( Ψ ) = h I | V ( i ) | Ψ i , (2.33)where V ( i ) is a closed string vertex operator, such as c ( i ) c ( − i ) V m ( i, − i ), where V m ( z, ¯ z ) is amatter primary with the conformal dimension (1 , h ( ± i ) = 0, in spite of theclosed string vertex on I , ˜ q ( h ) satisfies h I | V ( i ) ˜ q ( h ) = 0 . (2.34)In addition, the operators K n generate a global symmetry of the open string field theoryeven if the gauge invariant overlaps are introduced as sources. In fact, since V ( i ) has thedimension 0 and f ( ± i ) = ± i , we find that U f V ( i ) U − f = V ( f ( i )) = V ( i ) and then h I | V ( i ) U − f = h I | V ( i ) . (2.35)Consequently, we can see that the gauge invariant overlaps for the solution (2.12) are equiv-alent to that for the conventional solution (2.11): O V ( Φ ( K ′ , B, c )) = O V ( Ψ ( K, B, c )) . (2.36)10.6. Observables around the tachyon vacuum solution
Now let us consider observables for the classical solution (2.13) around the identity-basedtachyon vacuum solution. In this vacuum, the modified K ′ Bc algebra and the classicalsolution are simplified as mentioned before, and Q ′ has vanishing cohomology. From Q ′ c = 0 in (2.8), c turns out to be an exact state with respect to the modified BRSToperator Q ′ . Since Q ′ K ′ = 0, the solution (2.13) can be written as a modified BRST exactstate: Φ ( K ′ , c ) = Q ′ χ. (2.37)Therefore, we conclude that both the vacuum energy and the gauge invariant overlaps arezero for the classical solution (2.13).Here, we should note that the derivation of (2.37) requires careful consideration. InRef. 7), the homotopy operator is given for the BRST operator Q ′ in the identity-basedtachyon vacuum. In the case of the tachyon vacuum solution using the function (1.4) with a = − /
2, a corresponding homotopy operator is ˆ A = ( b (1) + b ( − /
2. If it is used forthe above exact form, such as χ = F ( K ′ ) ˆ Ac , the divergence arises from a collision between b (1) and c (1) and so we need regularization of (2.13). However, it should be noted thatthe homotopy operator ˆ A , such as { ˆ A, Q ′ } = 1, is not unique because we can add to it acommutator [ Q ′ , O ], where O is an arbitrary operator with ghost number −
2. Then, we havethe possibility of providing a regularization procedure by adding such terms to the homotopyoperator. In Ref. 17), several regularization methods are rigorously discussed.In addition, we should notice that the divergence does not appear in the procedure usedto analyze the cohomology of Q ′ in Ref. 2). The string field c can be expanded in the Fockspace for each L -level and the lowest-level state in c is c | i : c = 12 π c | i + · · · . (2.38)For the identity-based tachyon vacuum using the function (1.4) with a = − / Q ′ has anoscillator expression: Q ′ = R + R + R − , (2.39)where R n stands for terms with the mode number n with respect to L : ∗ ) R ± = − Q ± + c ± , R = 12 Q B + 2 c . (2.40) ∗ ) Here, we have expanded the conventional primary BRST current j B as j B ( z ) = P ∞ n = −∞ Q n z − n − and therefore Q = Q B in particular. The nilpotency of Q ′ leads to the anticommutation relations, { R ± , R ± } = 0 , { R ± , R } = 0 , { R , R − } + { R , R } = 0 . Q ′ c is written by the Fock space state starting from a lowest-level state: Q ′ c = 12 π R c | i + · · · . (2.41)Therefore, we can solve the equation Q ′ c = 0 level by level and then c can be given by amodified BRST exact state with no divergence, because, as in Ref. 2), the cohomology of Q ′ is expressed by the well-defined Fock space expression. In particular, there is no cohomologywithin the ghost number one sector. Thus, by solving the cohomology level by level, wecan write c as a Q ′ exact state with a well-defined Fock space expression. As a result, theexpression (2.37) can be well-defined with no divergence.2.7. Comments on a simplified algebra
Here, we comment on the case that e h ( z ) for the identity-based solution has zeros on theunit circle but not at z = ±
1. In Ref. 2), the identity-based solutions (1.1) with the function h la ( z ) ( l = 1 , , , · · · ; a ≥ − / h la ( z ) = log (cid:16) − a − l (cid:0) z l − ( − z − ) l (cid:1) (cid:17) (2.42)were considered as a generalization of the function h a ( z ) (1.4), which is the case of l = 1 inthe above. The solution corresponding to h la ( z ) is pure gauge for a > − / a = − /
2, the correspondingsolution is believed to represent the tachyon vacuum, where the BRST operator Q ′ aroundthe solution has no cohomology, and we have e h l − / ( z ) = ( − l z l + ( − z − ) l ) . (2.43)It has zeros at z = ± l is a positive odd integer and we can use simplified algebra(2.8) in the same way as the case of the function (1.4).In the case that l = 2 m ( m = 1 , , · · · , ), i.e. a positive even integer, the function (2.43)has zeros on the unit circle: z k = e iθ k , where θ k = k − m π ( k = 1 , , · · · , m ), and they arenot ±
1. In this case, the simplified K ′ Bc algebra (2.8) does not hold because e h m − / (1) = 0.However, we can obtain a simplified algebra by using c ′ = 1 π cos θ e − iθ c ( e iθ ) I (2.44)with θ = π m instead of c = π c (1) I as follows. Firstly, we note that e αK c = 2 π U † U ˜ c ( α ) | i , (2.45)12here ˜ c (˜ z ) = tan ◦ c (˜ z ) and we have used the notation in Ref. 6). Using a relation( U † U ) − c ( e iθ )( U † U ) = (cid:16) cos( it + π (cid:17) − ˜ c ( it ) , e iθ = tan( it + π , (2.46)we have e itK c = 2 π cos ( it + π c ( e iθ ) I = 1 π cos θ e − iθ c ( e iθ ) I. (2.47)Therefore, with t such as e iθ = tan( it + π ), or t = arctanh(tan θ ), c ′ defined in (2.44)can be expressed as c ′ = e it K c . Because K = L + L − is a derivation with respect to thestar product, and noting [ K , ( B ) L ] = 0 and [ K , Q B ] = 0, we have Q B c ′ = e it K Q B c = c ′ Kc ′ , Bc ′ + c ′ B = e it K ( Bc + cB ) = 1 , ( c ′ ) = e it K c = 0 , (2.48)and they form a kind of KBc ′ algebra. Furthermore, noting (2.6) and (2.7), we obtain asimplified algebra K ′ = Q ′ B, Q ′ K ′ = 0 , Q ′ c ′ = 0 , (2.49) B = 0 , ( c ′ ) = 0 , Bc ′ + c ′ B = 1 , (2.50)for a modified BRST operator Q ′ corresponding to the function h m − / ( z ). Using the above,we can apply the prescription in Sect. 2.6 in a similar way. §
3. Observables for identity-based solutions
We consider direct calculation of observables for the identity-based tachyon vacuum so-lutions, by use of the method for the identity-based marginal solution in Ref. 11).We consider one parameter family of the identity-based solution, Ψ ( a ). The parameter a deforms the function h ( z ) in the solution, and as the simplest case (1.4) it takes values a ≥ − /
2, the solution becomes the tachyon vacuum at a = − / ∗ ) otherwise it is a trivialsolution. In particular, we assume that Ψ ( a = 0) = 0.Suppose that Ψ ( K, B, c ) in (2.11) is a tachyon vacuum solution in the conventionaltheory. Then, Φ ( K ′ , B, c ) in (2.12) is a tachyon vacuum solution in the theory with Q ′ for a > − /
2, but, in the case of a = − / Φ ( K ′ , c ) in (2.13) is a trivial pure gauge solution.Here, we take Ψ a = Ψ ( a ) + Φ for a ≥ − /
2. We can easily find that Ψ a is a classicalsolution in the conventional theory, namely it satisfies Q B Ψ a + Ψ a = 0. Expanding the stringfield around Ψ a in the action, we have the kinetic operator Q Ψ a : Q Ψ a A = Q B A + Ψ a A − ( − | A | AΨ a for all string field A . Q Ψ a can be written as Q Ψ a A = Q ′ A + Φ A − ( − | A | AΦ = Q ′ Φ A, (3.1) ∗ ) Namely, e h ( z ) has zeros on the unit circle at a = − / Q ′ is the modified BRST operator in (1.3) and Q ′ Φ represents the kinetic operatoraround the solution Φ in the theory at the identity-based vacuum Ψ ( a ). The importantpoint is that we can construct a homotopy operator for Q ′ Φ (= Q Ψ a ) for a ≥ − / K ′ Bc algebra. ∗ ) Differentiating the equation of motion, Q B Ψ a + Ψ a = 0, with respect to a , we find Q Ψ a dda Ψ a = 0 . (3.2)Since Q Ψ a has vanishing cohomology, we have dda Ψ a = Q Ψ a Λ a , (3.3)for some state Λ a . Integrating (3.3) from a = 0, we get Ψ ( a ) + Φ = Ψ ( K, B, c ) + Z a Q Ψ a Λ a da, (3.4)where we have used the fact that in the case of a = 0, Ψ ( a = 0) = 0 and the K ′ Bc solutionis the same as the conventional tachyon vacuum solution: Φ ( K ′ , B, c ) = Ψ ( K, B, c ).From (3.4), we can calculate the gauge invariant overlap for the identity-based solution: O V ( Ψ ( a )) = O V ( Ψ ( K, B, c )) − O V ( Φ ) , (3.5)where we have used the fact that the gauge invariant overlap is BRST invariant with respectto Q Ψ a : O V ( Q Ψ a ( · · · )) = 0. Noting that the formula (3.5) holds for a ≥ − /
2, by using theresult of the gauge invariant overlap for Φ in the previous section, the above is evaluated as O V ( Ψ ( a )) = a > − / π D V ( i ∞ ) c ( π E C π ( a = − / , (3.6)where we have used the notation in Ref. 11). Thus, as expected for the identity-basedsolution, the gauge invariant overlap for a > − / a = − /
2, the gauge invariant overlap agrees with the resultfor the tachyon vacuum solution.As emphasized in Ref. 12), the formula (3.4) is nothing but a gauge equivalence relationbetween Ψ ( a ) + Φ and Ψ ( K, B, c ). In fact, given the relation (3.4), Ψ ( a ) + Φ can bewritten as Ψ ( a ) + Φ = g − Q B g + g − Ψ ( K, B, c ) g, (3.7) ∗ ) In the case that Ψ ( K, B, c ) is the Erler-Schnabl solution, e.g., we have Φ ( K ′ , B, c ) = √ K ′ ( c + cK ′ Bc ) √ K ′ and the corresponding homotopy operator is given by a homotopy state: √ K ′ B √ K ′ inthe same way as Ref. 13). g is given by the path-ordered exponential form, g = P exp (cid:18)Z a Λ a da (cid:19) . (3.8)From this gauge equivalence relation, we have S [ Ψ ( a ); Q B ] + S [ Φ ; Q ′ ] = S [ Ψ ( K, B, c ); Q B ] . (3.9)From the result for S [ Φ ; Q ′ ] in the previous section and for the conventional tachyon vacuum,namely S [ Ψ ( K, B, c ); Q B ] = 1 / (2 π ), we finally find that − S [ Ψ ( a ); Q B ] = a > − / − π ( a = − / . (3.10)Thus, we have evaluated the vacuum energy density for the identity-based solution and theseresults are consistent with our expectation for the solution. §
4. Concluding remarks
We have constructed classical solutions Φ in the theory expanded around the identity-based scalar solution Ψ (1.1). We have taken advantage of the K ′ Bc algebra to calculateobservables for the solution. In the case that Ψ is trivial pure gauge, the observablesfor Φ are equivalent to those for the corresponding solution Ψ ( K, B, c ) in the originalbackground. In the case that Ψ is the tachyon vacuum, they become equal to those fortrivial solutions, since the K ′ Bc algebra is simplified and all the solutions made from K ′ , B ,and c are given as Q ′ -exact states. Finally, we have provided the gauge equivalence relationbetween Ψ ( a ) + Φ and Ψ ( K, B, c ), which is regarded as a new expression for the identity-based solution. Thanks to this expression, we have analytically calculated observables forthe identity-based scalar solution whether it corresponds to trivial pure gauge or tachyonvacuum.Around the identity-based tachyon vacuum solution, the zeros of e h ( z ) on the unit circleplay a crucial role in evaluating observables for Φ . As seen in Sect. 2.7, there is no need forthese zeros to be at z = ±
1, which correspond to open string boundaries. We note that wecan find similar results in the study of homotopy operators for the BRST operator aroundthe identity-based scalar solutions, in which homotopy operators exist only if the zerosare on the unit circle. Here, we should comment on another identity-based solution discussedin Ref. 3), in which e h ( z ) has higher-order zeros than the function in this paper. However,15imilar to the discussion of homotopy operators in Ref. 7), we can obtain the simplified K ′ Bc algebra with higher-order zeros and an important point is the position of the zeros ratherthan the order.For the simplest function h a ( z ) (1.4), the solution Φ ( K ′ , B, c ) in the theory around Ψ ( a )depends on the parameter a . We find that for a > − / Φ can correspond to the tachyonvacuum but for a = − /
2, it becomes a trivial pure gauge configuration as stated in Sect. 2.6.This result is in accordance with the numerical analysis in Ref. 24), where it is observed thatin the theory around Ψ ( a > − ), we can construct a numerical solution whose energydensity corresponds to the negative of the D-brane tension, while the solution continuouslyconnects to trivially zero as a approaches − /
2. We find that the transition becomes sharpif the truncation level increases. Accordingly, Φ can be identified as the numerical solutionin Ref. 24) although they belong to different gauge sectors. Here, we should note that, for a = − /
2, we expect that there exists a solution whose energy density is the positive of theD-brane tension. Such a solution has been constructed numerically in the Siegel gauge inRefs. 9), 10). It should represent the perturbative vacuum where a D-brane exists. ∗ ) Acknowledgements
We would like to thank Loriano Bonora and other organizers of the conference SFT2014 atSISSA, Trieste for the kind hospitality. We are grateful to Nobuyuki Ishibashi for productivediscussions. The work of the authors is supported by a JSPS Grant-in-Aid for ScientificResearch (B) (
Appendix A
Solutions for the differential equation (2.18)
Let us consider solutions of (2.18). It is sufficient to solve the equation in an annulusincluding the unit circle | z | = 1, because now we look for a regular function f ( z ) on thecircle.The differential equation (2.18) is reducible to the homogeneous equation, dg ( z ) dz − iX ( z ) g ( z ) = 0 , X ( z ) = − z ) e h ( z ) , (A.1) ∗ ) N. Ishibashi pointed out the possibility of constructing the perturbative vacuum solution in a privatediscussion.
16y the variable transformation, f ( z ) = i g ( z ) + i g ( z ) − i . (A.2)Solving Eq. (A.1) under the initial condition f (1) = 1, the function f ( z ) is given by f ( z ) = − i ie v ( z ) − ie v ( z ) , v ( z ) = 2 i Z z X ( z ′ ) dz ′ . (A.3)Since the function X ( z ) has singularity at z = ± i , v ( z ) becomes divergent and so theexpression (A.3) is undefined at the midpoints. Here, let us analyze the behavior of f ( z )near the midpoints in terms of series solutions. Suppose that h ( z ) is holomorphic at z = i .Since X ( z ) has a single pole at z = i , X ( z ) is expanded into a Laurent series: X ( z ) = 1 z − i ∞ X n =0 x n ( z − i ) n , (A.4)where the first few coefficients are given by x = ie − h ( i ) , x = − e − h ( i ) i∂h ( i )) , · · · . (A.5)Using this expansion, we can construct a series solution for the differential equation (A.1): g ( z ) = ( z − i ) λ ∞ X n =0 A n ( z − i ) n , (A.6)where we find that A = 0, λ = − e − h ( i ) , and other A n are given by a recurrence formula: A n = 2 in ( x A n − + x A n − + · · · + x n A ) . (A.7)It can be easily seen that this series solution is convergent in a neighborhood of z = i . Since λ = − h ( ± i ) = 0 for the identity-based solution, f ( z ) is a holomorphic functionnear z = i : f ( z ) = i z − i ) g ( z ) + i ( z − i )2( z − i ) g ( z ) − i ( z − i ) . (A.8)Taking the limit z → i , we find that f ( i ) = iA /A = i . In addition, we have f ′ ( i ) = − /A = 0. Similarly, we find that f ( z ) is holomorphic at z = − i , and that f ( − i ) = − i and f ′ ( − i ) = 0.We have found that the poles of X ( z ) at z = ± i are harmless to solve (2.18). However, if X ( z ) has poles on the unit circle due to zeros of e h ( z ) , it is difficult to find regular solutionson the unit circle. Suppose that X ( z ) is expanded around the zero z ( = ± i, | z | = 1) as X ( z ) = x ′ z − z + x ′ + · · · , (A.9)17 ( z ) behaves around z = z as g ( z ) ∼ ( z − z ) ix ′ × ( · · · ) , (A.10)where the dots denote a regular function. Here it is noted that X ( z ) has poles on the unitcircle, but the residue x ′ is essentially unrestricted as opposed to the residue x at z = ± i .Therefore, g ( z ) is not a regular function in general and so it is impossible to find a regularconformal transformation f ( z ) if e h ( z ) has zeros on the unit circle. Actually, we have seenan example for a singular map in Sect. 2.4.Now, let us consider (2.19) and (2.20) for the solution (A.3). For z = e iσ , ( | z | = 1), v ( z )is written by v ( e iσ ) = Z σ e h ( e iσ ) cos σ dσ. (A.11)As mentioned in the introduction, the reality condition of Ψ implies ( h ( z )) ∗ = h (1 /z ∗ ) andso, for z = e iσ , h ( z ) is a real-valued function. Then, from (A.11), we find that v ( e iσ ) isreal-valued. Consequently, from (A.3), we find that | f ( z ) | = 1 for | z | = 1.For z = e iσ , we write the phase of f ( z ) as φ ( σ ): φ ( σ ) = 1 i ln f ( e iσ ) . (A.12)Differentiating the phase with respect to σ , we have dφ ( σ ) dσ = 2 e v ( e iσ ) { e e v ( eiσ ) } e h ( e iσ ) cos σ . (A.13)Since v ( e iσ ) and h ( e iσ ) are real, the derivative is positive for | σ | ≤ π/ φ ( σ ) is amonotonically increasing function from − π/ π/
2. Hence, we have proved the properties(2.19) and (2.20).Finally, we consider the inversion formula (2.21). We note that the differential equation(2.18) has symmetries under the following transformations: z → − z , (A.14) f → af + b − bf + a , ( a + b = 1 , a, b ∈ C ) . (A.15)The first is a Z transformation derived from h ( − /z ) = h ( z ), which is needed for theidentity-based solution as mentioned in the introduction. The second transformation formsthe group SO (2 , C ) in which f = ± i are fixed points. Therefore, if a special solution f ( z )is known, a general solution is given by the above SO (2 , C ) transformation of f ( z ). In18act, SO (2 , C ) has two real parameters and these correspond to integration constants for thecomplex first-order differential equation (2.18). Then, since f ( − /z ) is also a solution dueto the first symmetry, we find that the relation f (cid:18) − z (cid:19) = af ( z ) + b − bf ( z ) + a (A.16)has to hold for some SO (2 , C ) parameters a, b . By performing this transformation twice,we can determine the parameters as ( a, b ) = (1 ,
0) or (0 , f ( z ) isholomorphic at z = i and f ′ ( i ) = 0, f ( z ) must satisfy the inversion formula (2.21). References
1) T. Takahashi and S. Tanimoto, “Marginal and scalar solutions in cubic open stringfield theory,” JHEP , 033 (2002) [hep-th/0202133].2) I. Kishimoto and T. Takahashi, “Open string field theory around universal solutions,”Prog. Theor. Phys. , 591 (2002) [hep-th/0205275].3) Y. Igarashi, K. Itoh, F. Katsumata, T. Takahashi and S. Zeze, “Classical solutionsand order of zeros in open string field theory,” Prog. Theor. Phys. , 695 (2005)[hep-th/0502042].4) E. Witten, “Noncommutative Geometry and String Field Theory,” Nucl. Phys. B , 253 (1986).5) Y. Okawa, “Comments on Schnabl’s analytic solution for tachyon condensation inWitten’s open string field theory,” JHEP , 055 (2006) [hep-th/0603159].6) M. Schnabl, “Analytic solution for tachyon condensation in open string field theory,”Adv. Theor. Math. Phys. , 433 (2006) [hep-th/0511286].7) S. Inatomi, I. Kishimoto and T. Takahashi, “Homotopy Operators and One-LoopVacuum Energy at the Tachyon Vacuum,” Prog. Theor. Phys. , 1077 (2011)[arXiv:1106.5314 [hep-th]].8) T. Takahashi and S. Zeze, “Gauge fixing and scattering amplitudes in stringfield theory around universal solutions,” Prog. Theor. Phys. , 159 (2003)[hep-th/0304261].9) I. Kishimoto and T. Takahashi, “Vacuum structure around identity based solutions,”Prog. Theor. Phys. , 385 (2009) [arXiv:0904.1095 [hep-th]].10) I. Kishimoto, “On numerical solutions in open string field theory,” Prog. Theor.Phys. Suppl. , 155 (2011).11) I. Kishimoto and T. Takahashi, “Gauge Invariant Overlaps for Identity-Based19arginal Solutions,” Prog. Theor. Exp. Phys. , (2013) [arXiv:1307.1203[hep-th]].12) I. Kishimoto and T. Takahashi, “Comments on observables for identity-basedmarginal solutions in Berkovits’ superstring field theory,” JHEP , 031 (2014)[arXiv:1404.4427 [hep-th]].13) S. Inatomi, I. Kishimoto and T. Takahashi, “Tachyon Vacuum of Bosonic OpenString Field Theory in Marginally Deformed Backgrounds,” PTEP , 023B02(2013) [arXiv:1209.4712 [hep-th]].14) S. Inatomi, I. Kishimoto and T. Takahashi, “On Nontrivial Solutions around aMarginal Solution in Cubic Superstring Field Theory,” JHEP , 071 (2012)[arXiv:1209.6107 [hep-th]].15) C. Maccaferri, “A simple solution for marginal deformations in open string fieldtheory,” JHEP , 004 (2014) [arXiv:1402.3546 [hep-th]].16) M. Kiermaier, Y. Okawa and P. Soler, “Solutions from boundary condition changingoperators in open string field theory,” JHEP , 122 (2011) [arXiv:1009.6185[hep-th]].17) N. Ishibashi, “Comments on Takahashi-Tanimoto’s scalar solution,” arXiv:1408.6319[hep-th].18) S. Zeze, “Gauge invariant observables from Takahashi-Tanimoto scalar solutions inopen string field theory,” arXiv:1408.1804 [hep-th].19) B. Zwiebach, “Interpolating string field theories,” Mod. Phys. Lett. A , 1079 (1992)[hep-th/9202015].20) I. Ellwood, “The Closed string tadpole in open string field theory,” JHEP , 063(2008) [arXiv:0804.1131 [hep-th]].21) 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 [hep-th]].22) T. Erler and M. Schnabl, “A Simple Analytic Solution for Tachyon Condensation,”JHEP , 066 (2009) [arXiv:0906.0979 [hep-th]].23) S. Inatomi, I. Kishimoto and T. Takahashi, “Homotopy Operators and Identity-Based Solutions in Cubic Superstring Field Theory,” JHEP , 114 (2011)[arXiv:1109.2406 [hep-th]].24) T. Takahashi, “Tachyon condensation and universal solutions in string field theory,”Nucl. Phys. B670