$KBc$ algebra and the gauge invariant overlap in open string field theory

We study in detail the evaluation of the gauge invariant overlap for analytic solutions constructed out of elements in the $KBc$ algebra in open string field theory. We compute this gauge invariant observable using analytical and numerical techniques based on the sliver frame $\mathcal{L}_0$ and traditional Virasoro $L_0$ level expansions of the solutions.


Introduction
It is well-known that the analytic solutions for tachyon condensation [1,2,3,4,5,6,7] in open bosonic string field theory [8] can be formally given in terms of elements in the KBc algebra [9,10]. Once a solution Ψ is given, the next step is to evaluate relevant physical gauge invariant quantities, like the energy and the gauge invariant overlap I|V(i)|Ψ discovered in [11,12,13]. As argued by Ellwood [14], the gauge invariant overlap represents the shift in the closed string tadpole of the solution relative to the perturbative vacuum. Moreover, using an appropriate zero momentum vertex operator V, defined in [15], it has been shown that the value of the energy can be obtained from the gauge invariant overlap.
The analytic computation of the gauge invariant overlap for Schnabl's tachyon vacuum solution has been performed in [13]. Although the evaluation of this gauge invariant appears to be simpler than the energy, the computation presented in [13] was a bit cumbersome, and the reason for this subtlety was that the authors used a representation of the solution as given in Schnabl's original work [1]. As we will see, the computation of the gauge invariant overlap can be enormously simplified if we express Schnabl's solution in terms of elements in the KBc algebra.
Concerning the numerical analysis of the gauge invariant overlap for analytic solutions within the KBc algebra, in reference [13], using the traditional Virasoro L 0 level truncation scheme the authors have evaluated the gauge invariant overlap for the case of Schnabl's original solution. Regarding the case of the so-called Erler-Schnabl's solution, although the analytical computation of the gauge invariant overlap for this solution has been performed in [2], up to now, using the Virasoro L 0 level truncation scheme, the Under the map (2.1), the UHP looks as a semi-infinite cylinder of circumference π denoted by C π .
There is another convention for the definition of the sliver frame which uses the following mapz = 2 π arctan z. (2.2) This map has been used in reference [2], and in this case, the UHP looks as a semi-infinite cylinder of circumference 2 denoted by C 2 .
Since the expressions written in terms of elements in the KBc algebra which are used in the construction of analytic solutions look different depending on the convention adopted for thez coordinate, it is always useful to mention, from the beginning, what of those conventions will be chosen, i.e., the one given by (2.1) or (2.2).
In the literature some authors use the convention (2.1) and others (2.2), in this work we are going to use a rather generic definition which takes into account these two conventions. Let us define thez coordinate by the map z = l π arctan z, (2.3) so that the UHP looks as a semi-infinite cylinder of circumference l denoted by C l . Note that the case l = π corresponds to the convention (2.1) while the case l = 2 corresponds to (2.2).
Let us define the operatorsL,B andc p which are very useful in the construction of elements in the KBc algebra. These operators are related to the worldsheet energymomentum tensor T , the b and c ghosts fields respectively. Using the map (2.3), we can write the explicit definition of the operatorsL,B andc p L ≡ L 0 + L † 0 = dz 2πi (1 + z 2 )(arctan z + arccotz) T (z) , (2.4) In general, if we have a primary field φ with conformal weight h, using the map (2.3), we obtainφ (2.7) Using this equation (2.7), let us define the operators L −1 and B −1 which are useful in the computation of the star product of string fields involving the operatorsL andB To compute the star product of string fields involving the operatorsL,B andc p , we will need to know the following commutator and anti-commutator relations (2.10) To represent the elements in the KBc algebra, we will require to know the operator U † r U r . This operator can be written in terms of the operatorL 3 Star products and the KBc algebra Before defining the basic elements belonging to the KBc algebra, we are going to write the star product of string fields containing the operatorsL andB. Given two string fields φ 1 and φ 2 , we can show that where gn(φ) takes into account the Grassmannality of the string field φ. If we set l = π, the above results match the results given in reference [1].
The action of the BRST, L −1 , and B −1 operators on the star product of two string fields are given by Given a operatorφ(z) defined in thez coordinate, let us write the wedge state with insertion as follows where U r = (2/r) L 0 is the scaling operator in thez coordinate. The star product of two states U † r U rφ (x)|0 and U † s U sψ (ỹ)|0 can be derived using the usual gluing prescription where byφ(z) we have denoted the local operator φ(z) expressed in the sliver frame. For instance, in the case of a primary field with conformal weight h,φ(z) is given bỹ The elements in the KBc algebra are constructed out of the basic string fields K, B, and c. These fields can be represented in terms of operators acting on the identity string Let us derive the algebra associated to the set of operators defined by equations (3.13)-(3.15). As a pedagogical illustration, we explicitly compute {B, c} using equations (3.1), (3.2) and the anti-commutator (2.11), we obtain therefore, we have that {B, c} = 1.
Following the same steps, using equations (3.1)-(3.6), the commutator and anti-commutator relations (2.10), (2.11), we can show that (3.18) where the expression ∂c has been defined as ∂c ≡ U † 1 U 1 ∂c(0)|0 . The action of the BRST operator Q on the basic string fields K, B, and c is given by Employing the elements in the KBc algebra, we can construct a rather generic solution which formally satisfies the string field equation of motion QΨ+ΨΨ = 0. For this solution to be a well defined string field, the function F (K) must satisfy some holomorphicity conditions stated in reference [3]. From now, we will assume that Ψ belongs to the set of well defined string fields.
Let us list some solutions of the form (3.20). As a first example, consider the analytic solution for the tachyon vacuum [1], where F (K) = e −lK/4 , actually Schnabl's original solution corresponds to the case where l = π. Recall that in this work, we are considering the mapz = l π arctan z, and therefore the Schnabl's solution looks like There is a subtlety with this solution, as shown in references [1,3] when one performs the expansion of K/(1 − e −lK/2 ) as the sum n Ke −lKn/2 , the truncation of this sum produces a remnant which still contributes to certain observable [4]. This is the origin of the phantom term Ψ N . Taking into account the phantom term, the solution (3.21) can be written as where ψ n = e −lK/4 cBe −lKn/2 ce −lK/4 . (3.23) As a second example, let us consider the solution discovered by Erler and Schnabl, namely, the so-called simple tachyon vacuum solution [2] Ψ Er-Sch = 1 Note that in this case, F (K) = 1/ √ 1 + K, and as shown in references [2,3] there is no need for a phantom like term. It is possible to provide an integral representation of the solution (3.24), this is given by writing the inverse square root of 1 + K as where Ω t is the wedge state which can be written as [18,19] And as a last example, we consider the so-called, real tachyon vacuum solution without square roots, or Jokel's real solution for short [6,7]. This solution takes the form where the Q-exact terms are given by Interestingly, the solution does not take the factorized form (3.20), and is both real and simple, namely without square roots and phantom terms. For this real solution, the corresponding energy has been computed and shown that the value is in agreement with the value predicted by Sen's conjecture.

The gauge invariant overlap: analytical computations
In this section, we are going to study the analytic computation of the gauge invariant overlap for solutions given in terms of elements in the KBc algebra. This gauge invariant observable has been considered in references [11,12,13,15,20,21]. For a given solution Ψ of the string field equations of motion, the gauge invariant overlap is defined as the evaluation of the quantity where |I is the identity string field, and the operator V(i) is an on-shell closed string vertex operator V = ccV m1 which is inserted at the midpoint of the string field Ψ. As argued by Ellwood [14], the gauge invariant overlap represents the shift in the closed string tadpole of the solution relative to the perturbative vacuum.
To evaluate the gauge invariant overlap for solutions given in terms of elements in the KBc algebra, it will be useful the following results where the coefficient C V represents the correlator which is the closed string tadpole evaluate on a cylinder C 1 of unit circumference. The proofs of the above results (4.2)-(4.4) are based on usual scaling arguments and can be found in references [2,22].
As an application of equations (4.2)-(4.4), we are going to compute the gauge invariant overlap for Schnabl's tachyon vacuum solution. We would like to mention that in reference [13], after performing lengthy computations the authors have evaluated the gauge invariant overlap for Schnabl's solution. However, as we will see, this computation can be performed in a few lines if one uses Schnabl's solution expressed in terms of the basic string fields K, B and c therefore, we need to compute V|ψ n . Using equation (3.23), we can write Employing equation (4.4), from equation (4.7), we get plugging this result (4.8) into equation (4.6), we obtain This result coincides with the expected answer of closed string tadpole on the disk [14]. Note that the result (4.9) does not depend on the parameter l which explicitly appears in the solution (3.22).
Next we would like to evaluate the gauge invariant overlap for Erler-Schnabl's solution. Actually, using a non-real version of the solution (3.24), the computation of the gauge invariant overlap has been performed in reference [2]. Here we are going to present the computation for the case of the real solution 2 . Let us write the real solution (3.24) as the following integral representation , (4.10) therefore the gauge invariant overlap for this solution (4.10) will be given by . (4.11) Employing equation (4.4), from equation (4.11), we write As we can see, this result (4.12) is exactly the same as the one obtained for Schnabl's solution (4.9).
And as the last example of analytical calculation, let us evaluate the gauge invariant overlap for Jokel's real solution. Since BRST exact terms do not contribute for the evaluation of the gauge invariant overlap, we just need to consider the non-BRST exact terms of the solution. These terms are given on the right hand side of equation (3.27) and they can be written aŝ Therefore the gauge invariant overlap for Jokel's real solution is given by (4.14) Using equations (4.2) and (4.4), from equation (4.14) we obtain Note that this result (4.15) is the same as the ones obtained in the case of Schnabl's (4.9) and Erler-Schnabl's solutions (4.12).
It should be nice to obtain the above analytic results by numerical means. For instance, using the traditional Virasoro L 0 level truncation scheme, in reference [13], the authors have evaluated the gauge invariant overlap for Schnabl's solution. However, up to now, using the Virasoro L 0 level truncation scheme, the analysis of the gauge invariant overlap for Erler-Schnabl's and Jokel's real solution was not performed. Moreover, the analysis of the gauge invariant overlap by means of the curly L 0 level truncation scheme has not been carried out neither for Schnabl's, Erler-Schnabl's nor for Jokel's real solution.
In the next two sections, using the curly L 0 and the Virasoro L 0 level truncation scheme, we are going to present the evaluation of the gauge invariant overlap for solutions constructed out of elements in the KBc algebra.

The gauge invariant overlap: L 0 level truncation computations
Since from the beginning, we do not know if the result for the gauge invariant overlap obtained by analytical computations will match the result obtained by numerical means (either by using the L 0 or the L 0 level truncation scheme), it is important for the consistency of the solutions to explicitly check if these different schemes provide the same answer. In this section, using the L 0 level expansion of a rather generic solution Ψ, we will present the evaluation of the gauge invariant overlap.
As we know, the solution is given in terms of elements in the KBc algebra (which involves the operatorsL,B andc), in general, we can write the following L 0 level expansion where n = 0, 1, 2, · · · , and p, q = 1, 0, −1, −2, · · · . The coefficients of the expansion f n,p and f n,p,q can be regarded as generic ones, obviously these coefficients depend on the solution we choose. For instance, for the case of Schnabl's solution (3.21), these coefficients are given by where B m are the Bernoulli's numbers.
To compute the gauge invariant overlap for solutions expanded in terms of L 0 eigenstates, we start by replacing the string field Ψ with z L 0 Ψ, so that states in the L 0 level expansion will acquire different integer powers of z at different levels. And as usual, at the end, we will simply set z = 1.
Let us start with the evaluation of the gauge invariant overlap as a formal power series expansion in z. Plugging the expansion (5.1) into the definition of the gauge invariant overlap (4.1), we obtain As we can see, we need to compute V|L nc p |0 and V|L nBc pcq |0 . To evaluate these quantities, we require expressL nc p |0 andL nBc pcq |0 in terms of elements in the KBc algebra, for this purpose, it will be useful the following relations where Employing the above relations, we can writeL nc p |0 andL nBc pcq |0 in terms of elements in the KBc algebrâ (5.10) Now we are in position to evaluate the quantities V|L nc p |0 and V|L nBc pcq |0 . For instance, using equations (4.2) and (5.9), let us compute Performing similar calculations as the above, using equations (4.2), (4.3) and (5.10), we obtain V|BL nc pcq |0 = l δ n,0 δ p,0 δ q,1 − δ n,0 δ q,0 δ p,1 C V .
Finally, plugging the results (5.11) and (5.12) into the definition of the gauge invariant overlap (5.4), and setting z = 1, we get To compute the gauge invariant overlap for a solution expanded in terms of L 0 eigenstates (5.1), we only need to know the value of the first three coefficients appearing at levels z −1 and z 0 . Remarkably, this result (5.13) is simpler than the one obtained for the case of the energy. Evaluating the energy in the L 0 level expansion gives a very complicated non-convergent series, though the series can be resummed numerically by means of the so-called Padé approximants to give a good approximation to the brane tension [1,2,16].
Let us apply the general result (5.13) for some particular solutions such as the Schnabl's solution Ψ Sch . Using the explicit expressions of the coefficients (5.2) and (5.3) which appear in the L 0 level expansion of Schnabl's solution, from equation (5.13) we obtain This result does not depend on the parameter l and is the same result as the one obtained from analytic computations.
In the case of Erler-Schnabl's solution Ψ Er-Sch , using its integral representation (4.10), we can compute the first three coefficients appearing in the L 0 level expansion of the solution Therefore, plugging these results (5.16) into equation (5.13), we obtain This result also does not depend on the parameter l and is the same result as the one obtained for the case of Schnabl's solution.
In the case of Jokel's real solution, we can also calculate the curly L 0 level expansion of the non-BRST exact terms of the solution (4.13). The first three coefficients of this L 0 level expansion are given by  In this section, using the L 0 level truncation scheme, the evaluation of the gauge invariant overlap will be shown. Since the solution Ψ involves the operatorsL,B andc, we can write its L 0 level expansion as follows where n i , m j , s ≤ −2 and p, q = 1, 0, −1, −2, · · · . The L n 's are the ordinary Virasoro generators with zero central charge c = 0 of the total (i.e. matter and ghost) conformal field theory. For instance, Schnabl's solution (3.22), with l = π, expanded up to level two states is given by To compute the gauge invariant overlap by means of the L 0 level truncation scheme, it is clear that if we insert the expansion (6.1) into the definition of the gauge invariant overlap (4.1), we will need to evaluate the quantities As an illustration, suppose we need to calculate V|L n c p |0 . Since for n ≤ −2 the operator L n does not annihilate the vacuum |0 , and in order to apply the commutator (6.6), we must first express the operator L n in terms of annihilation operators. This can be achieved if we use the fact that the on-shell closed string state V = ccV m is invariant by the transformation generated by K n = L n − (−1) n L −n , namely, we have [13] V|L n = V|(−1) n L −n . (6.8) And now, since L −n |0 = 0 for n ≤ −2, we are able to compute V|L n c p |0 using the commutator (6.6) Let us comment that for the case of the operator b n which corresponds to the modes of the ghost field b, we have a similar result as the one given by equation (6.8) [11,12,13,23] V|b n = V|(−1) n b −n . (6.10) As we have seen, after the use of the commutation and anti-commutation relations (6.4)-(6.7), we can express the quantities (6.3) as linear combinations of terms like If we substitute equation (6.12) into equation (6.11), it is clear that we will need to evaluate the quantity V|c(z)|0 . Using equations (4.2) and (5.5), we can compute this quantity Therefore, employing equations (6.12) and (6.13), we obtain As a first example, let us compute the gauge invariant overlap for Schnabl's solution expanded up to level two states 15) where the values of the coefficients t ′ , u ′ , v ′ and w ′ are given in equation (6.2). Using the property that V|L −2 = V|L 2 and V|b −2 = V|b 2 , the evaluation of the gauge invariant overlap reads as follows We would like to compare this result (6.16) with the one obtained in reference [13], where Schnabl's solution has been expanded in a slightly different basis. Instead of considering the Virasoro generators L n with zero central charge, the authors have used the α n 's oscillators, for instance, up to level two states, they have written the expansion Then by using an explicit oscillator representation for the on-shell closed string state, which can be found in references [13,24], the gauge invariant overlap for the expanded Schnabl's solution (6.17) turns out to be [13]  Taking into account higher level states, we have performed the computation of the gauge invariant overlap for Schnabl's solution, and the results we have obtained with the normalization C V = 1/(2π) are in agreement with the ones presented in reference [13]. We can consider this agreement as a test for the method of computing the gauge invariant overlap based on the use of the equations (6.8), (6.10) and the commutation and anticommutation relations (6.4)-(6.7).
The advantage of this method compare to the one presented in [13], is that we do not require to use an explicit oscillator representation for the on-shell closed string state. The implication of this observation will be reflected in the simplification of the evaluation of the gauge invariant overlap. Recall that the L 0 level expansion of analytic solutions constructed out of elements in the KBc algebra, as presented in (6.1), is naively given in terms of the total (matter+ghost) Virasoro generators L n , the b n and c p modes, and since we do not require to use an explicit oscillator representation for the on-shell closed string state, using the expansion (6.1) we can directly evaluate the gauge invariant overlap without the necessity of reexpressing the expansion in terms of the α n 's oscillators (which will require an additional work).
Before to study the numerical evaluation of the gauge invariant overlap for the case of Erler-Schnabl's and Jokel's solutions, we would like to mention some motivations for doing this computation. Firstly, using the L 0 level truncation scheme, the numerical analysis of the gauge invariant overlap for Erler-Schnabl's and Jokel's solutions has not been carried out. This analysis should be crucial if we want to confirm the analytic result. However, the main motivation for performing such numerical computations is to see whether or not higher level contributions yield to increasingly convergent results which approach to the expected answer. In the case of Schnabl's solution, it has been shown that every time we increase the level of the truncated solution, the gauge invariant overlap converges to the expected analytical result without the necessity of using any regularization scheme such as Padé approximants [13].
Let us start with the L 0 level truncation analysis of the gauge invariant overlap for Erler-Schnabl's solution. To simplify the computations, it will be useful to write the solution (3.24) in the following way Inserting the solution (6.21) into the definition of the gauge invariant overlap, the BRST exact term does not contribute, and so we only need to consider the first term appearing on the right hand side of equation (6.21), let us denote this term as To compare the L 0 level expansion of the string field (6.22) with the one presented in reference [2], we choose the value of the parameter l, which appears in the definition of the map (2.3), as l = 2. The L 0 level expansion of the string field (6.22) can be obtained from the following result [2,25] where r and x are given by The operator U r is defined as U r ≡ · · · e u 10,r L −10 e u 8,r L −8 e u 6,r L −6 e u 4,r L −4 e u 2,r L −2 . (6.25) To find the coefficients u n,r appearing in the exponentials, we use ,r (f 6,u 6,r (f 8,u 8,r (f 10,u 10,r (· · · (f N,u N,r (z)) . . . ))))) , where the function f n,un,r (z) is given by By performing the change of variables where u ∈ [0, ∞) and η ∈ (−1, 1), we are going to numerically evaluate the double integrals coming from equation (6.23).
Employing the above results, let us write the string field (6.22), expanded up to level four states To evaluate the gauge invariant overlap using the L 0 level truncation scheme, first we perform the replacement Ψ (1) → z L 0 Ψ (1) and then using the resulting string field z L 0 Ψ (1) , we define The value of the gauge invariant overlap is obtained just by setting z = 1. As we can see, our problem has been reduced to the computation of quantities like V|L n 1 L n 2 · · · L n i c p |0 which can be evaluated using equations (6.4)-(6.7), (6.8) and (6.14).
As an example, plugging the level expansion (6.29) into the definition (6.30), we obtain If we set z = 1, from equation (6.31) we get about 89% of the expected result for the gauge invariant overlap (4.12). This result may appear good, however, considering the string field (6.22) expanded up to level twenty four states, we obtain about 116% of the expected result. This behavior is in contrast with the case of Schnabl's solution, where it has been shown that every time we increase the level of the truncated solution, the gauge invariant overlap converges to the expected analytical result [13]. Therefore, as we suspect, for the case of Erler-Schnabl's solution, by naively setting z = 1, we are obtaining a non-convergent result. Recall that in numerical L 0 level truncation computations a regularization procedure based on Padé approximants produces desired results for gauge invariant quantities like the energy [2]. Let us see if after applying Padé approximants, we can obtain the expected answer for the case of the gauge invariant overlap.
To obtain a Padé approximant of order P n 1+n (z) for the gauge invariant overlap, we will need to know the series expansion of (6.30) up to the order z 2n−1 . For the numerical evaluation, we have considered the string field Ψ (1) expanded up to level twenty four states, so that we obtain a series expansion for (6.30) truncated up to the order z 23 . The explicit expression for the gauge invariant overlap, truncated up to this order, is given by As an illustration of the numerical method based on Padé approximants, let us compute the value of the gauge invariant overlap using a Padé approximant of order P 4 1+4 (z). First, we express V|Ψ (1) (z) as the rational function P 4 1+4 (z) Expanding the right hand side of (6.33) around z = 0 up to seventh order in z and equating the coefficients of z −1 , z 0 , z 1 , z 2 , z 3 , z 4 , z 5 , z 6 , z 7 with the expansion (6.32), we get a system of algebraic equations for the unknown coefficients a 0 , a 1 , a 2 , a 3 , a Replacing the value of these coefficients inside the definition of P 4 1+4 (z) (6.33), and evaluating this at z = 1, we get the following value of the gauge invariant overlap The results of our calculations are summarized in table 6.1. As we can see, the value of the gauge invariant overlap evaluated using Padé approximants confirms the expected analytic result (4.12). Although the convergence to the expected answer gets quite slow, by considering higher level contributions, we will eventually reach to the right value of the gauge invariant overlap V|Ψ (1) → 1 C V . Table 6.1: The Padé approximation for the value of the gauge invariant overlap V|z L 0 Ψ (1) divided by C V and evaluated at z = 1. The third column shows the P n 1+n Padé approximation. In the last column, P 2n 1 represents a trivial approximation, a naively summed series. At each line, we have considered the string field expanded up to level 2n states.
Level Finally, let us show the L 0 level truncation analysis of the gauge invariant overlap for Jokel's solution. In order to expand the string field (4.13) in the state space of Virasoro L 0 eigenstates, we need to write this string field as follows [7] Ψ Jok = ∞ 0 dt e −t r sin 2 π 2r 2πr − r sin π r + π 16π 2 U r c − 2 tan πt 2r r + c 2 tan πt where r = 1 + t.
By writing the c ghost in terms of its modes c(z) = m c m /z m−1 and employing equations (6.25) and (6.37), the string fieldΨ Jok can be readily expanded and the individual coefficients can be numerically integrated. For instance, let us write the expansion ofΨ Jok up to level four stateŝ In order to compute the gauge invariant overlap using the L 0 level truncation scheme, we perform the replacementΨ Jok → z L 0Ψ Jok and then using the resulting string field z L 0Ψ Jok , we define It turns out that if we naively set z = 1 in (6.39), we obtain a non-convergent result, therefore in the case of Jokel's solution, we are also required to use Padé approximants.
We have considered the string fieldΨ Jok expanded up to level twenty four states, so that we obtain a series expansion for (6.39) truncated up to the order z 23 . The explicit expression for the gauge invariant overlap, truncated up to this order, is given by Starting from this expression (6.40), we have computed the value of the gauge invariant overlap using Padé approximants of order P n 1+n (z). Since these computations are similar to the ones developed in the case of Erler-Schnabl's solution, at this point we only present the results which are shown in table 6.2. We observed that the value of the gauge invariant overlap evaluated using Padé approximants confirms the expected analytic result. Table 6.2: The Padé approximation for the value of the gauge invariant overlap V|z L 0Ψ Jok divided by C V and evaluated at z = 1. The third column shows the P n 1+n Padé approximation. In the last column, P 2n 1 represents a trivial approximation, a naively summed series. At each line, we have considered the string field expanded up to level 2n states. Level

Summary and discussion
Through analytical and numerical techniques, we have evaluated the gauge invariant overlap for solutions within the KBc algebra. In order to numerically analyze the gauge invariant overlap, we have used two types of expansions for the truncated solutions, namely, the curly L 0 and the Virasoro L 0 level expansions.
We have shown that when we expand a solution Ψ in the basis of curly L 0 eigenstates, the resulting expression for the gauge invariant overlap I|V(i)|Ψ is given in terms of a finite series and so the use of Padé approximants was not necessary. This is a quite generic result provided that the solution belongs to the state space constructed out of elements in the KBc algebra. As explicit examples, we have presented the results for the case of Schnabl's, Erler-Schnabl's and Jokel's solutions.
Regarding the Virasoro L 0 level truncation analysis of the gauge invariant overlap for Erler-Schnabl's and Jokel's solutions, we have shown that the resulting expressions are given in terms of divergent series which nevertheless using Padé approximants can be numerically evaluated. These results are in contrast to the case of Schnabl's original solution where the expression of the gauge invariant overlap obtained from Virasoro L 0 level truncation computations becomes a convergent series, therefore, in that case [13], there was no need for using Padé approximants.
Our original motivation for studying the level truncation analysis of the gauge invariant overlap has been to prepare a numerical background to analyze more cumbersome solutions, such as the multibrane solutions [22], however there are problems that can arise when using the KBc algebra to construct such solutions, for instance, depending on the regularization used to define the solutions, the analytic computation of the energy and the gauge invariant overlap becomes ambiguous [26,27], moreover, these solutions are not well defined when expanded in the basis of Virasoro L 0 eigenstates [28].
With the hope of constructing well-behaved solutions other than the tachyon vacuum, recently, the KBc algebra has been extended to a larger algebra given as a string field representation of the Virasoro algebra [29]. Since the evaluation of the gauge invariant overlap is simpler than the energy, it should be nice to extend the results presented in our work in order to compute the gauge invariant overlap for solutions constructed within the proposed Mertes-Schnabl's algebra.
Finally, we would like to comment that the evaluation of the gauge invariant overlap can be generalized for solutions in the context of superstring field theory [30,31,32]. For instance, we should analyze the gauge invariant overlap for solutions constructed out of elements in the so-called GKBcγ algebra introduced in references [33,34,35,36,37]. The analytic computation of this gauge invariant quantity has already been presented for some particular solutions [38,39,40], however it remains the numerical analysis.