Observables for identity-based tachyon vacuum solutions

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. Introduction
An 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 identity-based 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 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 correct observables as expected.
The identity-based scalar solution is given by 1) Ψ 0 = Q L (e h − 1)I − C L ((∂h) 2 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) multiplied 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: where we denote the action as S[Ψ ; Q] = − 1 2 Ψ * QΨ + 1 3 Ψ * Ψ * Ψ and the kinetic oper-ator Q ′ is given by (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 it can be changed by gauge transformations. Since the function continuously connects to zero and so most of the solutions are regarded as a trivial pure gauge solution. However, nontrivial solutions is generated at the boundary of some function spaces. In the well-studied case, the function includes one parameter a ≥ −1/2: It is known that the solution for a > −1/2 is a trivial pure gauge, but it becomes a nontrivial solution for a = −1/2, for the reason mentioned above.
The transition from a trivial pure gauge to the tachyon vacuum solution has been observed on 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 when the zeros reach the unit circle |z| = 1. For example, e h(z) for (1 . 4) is rewritten as , (1 . 5) and it has zeros at ± −Z(a) and ±1/ −Z(a). When the parameter a approaches −1/2 from positive infinity, Z(a) runs from 1 to −1 and then it takes the value −1 for a = −1/2.
As a result, we find that the zeros are on the unit circle only if a = −1/2 and then the solution becomes the tachyon vacuum solution. 8) For other functions, we find the same transition occurs if the zeros move to the unit circle. 3) We now briefly outline our strategy. First, we find the KBc algebra in the shifted theory 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 is able to calculate observables for these solutions. Here, the shifted theory includes one parameter a, as the above example, through Q ′ , and so the classical solutions depend on the parameter. This is similar to the case of the analysis for the identity-based marginal solutions, 11), 12) in which the shifted theory and the solution include parameters related to marginal deformations. Therefore, according to the marginal case, we represent the identity-based solution as a gauge equivalence relation involving the identity-based and wedge-based solutions. 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 identitybased trivial solution and that around the identity-based non-trivial solution. If Ψ 0 is a trivial pure 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 solutions in the shifted theory. Here, the existence of such a transformation depends crucially on the positions of zeros of the function e h(z) . The zeros on the unit circle become obstacles to construction of the transformation and therefore it is impossible to transform from the K ′ Bc algebra to the KBc one in the case that Ψ 0 is the tachyon vacuum solution. However, it implies that, on the identity-based tachyon vacuum, K ′ , B and c have a different algebraic structure from the original KBc algebra. We will find that, on the identity-based tachyon vacuum, the operators K ′ and c commute with each other and then all the solutions made of K ′ , B and c can be written as modified BRST exact states. Accordingly, observables for them are calculable even if Ψ 0 is the tachyon vacuum solution.
This paper is organized as follows: First, we will consider classical solutions in the theory expanded around the identity-based solution in §2. We construct the K ′ Bc algebra with respect to Q ′ and, by using the K ′ Bc algebra, we will find classical solutions on the identity-based vacuum. To calculate observables for the classical solutions, we will construct a similarity transformation from the operator (K ′ 1 ) L to (K 1 ) L . We will find that a conformal transformation is a significant part for the similarity transformation and so we will illustrate it by an example for h a (z) in (1 . 4). Then, we will calculate observables for the classical solutions around Ψ 0 . In §3, based on the results in the previous section, we evaluate analytically observables for the identity-based solutions. In §4, we will give concluding remarks. In appendix A, we provide a detailed proof of properties of a differential equation which plays an important role on calculation of observables.
Note added: When we had a discussion with N. Ishibashi during the conference SFT2014 at SISSA, Trieste, it was found that we reached a same conclusion for the gauge invariant observables for the identity-based tachyon vacuum solution. * ) The main difference is that he argued in detail regularization method to evaluate the observables 17) but we evaluated the observables for the identity-based trivial solution in addition to the tachyon vacuum case.
After almost completing the manuscript, we found a paper by S. Zeze appearing on the arXiv, 18) which treats similar 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): where B and c are the same string fields in the conventional KBc algebra, 5) and K ′ is given by * * ) The operator (K ′ 1 ) L is explicitly calculated as 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 ′ 1 ) L equals to the conventional (K 1 ) L in the KBc algebra.
For the general function h(z), K ′ , B, c and Q ′ have the same algebraic structure as that of 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 the following equations: where K ′ 1 is the operator defined by the replacement of the integration path in (2 . 5) with a unit circle: As mentioned in §1, the function e h(z) has zeros * * ) We use following convention: 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 ≥ 0. * ) It can be calculated by using the relations in Ref . 7): on the unit circle in the case that the solution becomes the tachyon vacuum solution. Indeed, for (1 . 4), e ha(z) has zeros at z = ±1 only in the case a = −1/2 and then, from (2 . 6), Q(e ha ) and c(1) anticommutes with each other for a = −1/2. * ) Similarly, K ′ 1 and c(1) commutes for a = −1/2 from (2 . 7). Consequently, we find a simplified algebra only in the case that e h(z) has zeros at z = ±1, namely in the theory around the tachyon vacuum solution: Actually, there are other possibilities 2) where the function e h(z) for the tachyon vacuum solution has zeros on the unit circle but not at z = ±1. We will discuss these cases at the end of the section.

Classical solutions
The equation of motion in the theory around the solution (1 . 1) is given by where Q ′ is the modified BRST operator (1 . 3). We can find various classical solutions in the shifted background by substituting K ′ for K of the solutions given by the KBc algebra in the original theory. In the conventional theory with Q B , a classical solution using the KBc algebra is written as which is general configuration with ghost number one in terms of the KBc algebra. Here, 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 shifted 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 a simpler expression: and the second term in (2 . 12) vanishes due to K ′ c = cK ′ and c 2 = 0. * ) 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(−1)I.

2.3.
Transformations from (K ′ 1 ) L to (K 1 ) L In this subsection, we will consider a similarity transformation from (K ′ 1 ) L to the conventional (K 1 ) L .
First, we introduce the operator 1), 8) * ) where j gh (z) is the ghost number current, j gh = cb. Using this operator, the modified BRST operator (1 . 3) is transformed to the original BRST operator: 1) Accordingly, we can remove the ghost number current from (K ′ 1 ) L in (2 . 5) by a similarity transformation: Next, we look for a conformal transformation z ′ = f (z) which maps (2 . 16) to (K 1 ) L . Since T (z) is a primary field with the dimension 2, the operator (2 . 16) is transformed as where C ′ left is an integration path in the mapped plane such as f : C left → C ′ left . In order that (2 . 17) coincides with (K 1 ) L , the function f (z) must satisfy a differential equation: 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 an annulus including the unit circle |z| = 1. The important point is that we can solve it if e h(z) has no zeros on the unit circle as seen in appendix A. Moreover, we can prove that under the initial condition f (1) = 1, the solution f (z) has the following properties: .
We illustrate these by the solution given for h a (z) (1 . 4) in the next subsection and we give a detailed proof in appendix A. From (2 . 19) and (2 . 20), we find that the conformal map by the solution f leaves the integration path in (2 . 17) unchanged, namely C ′ left = C left . Therefore we can transform the operator (2 . 16) to (K 1 ) L by the conformal transformation f . Moreover, (2 . 21) indicates that Consequently, around the trivial pure gauge solution, we can construct the similarity where U f is the operator for the conformal transformation f and it is given in the form 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 this sense, we emphasize that a regular operator U f does not exist for the tachyon vacuum.

Observables around the trivial pure gauge solution
In this subsection, we will show that observables for the solution (2 . 12) around the identity-based trivial pure gauge solution are equivalent to those for the original solution (2 . 11).
First, we find that c(1) and (B 1 ) L are invariant under the similarity transformation (2 . 28) From the differential equation (2 . 18), it follows that and then c(1) is invariant under the transformation because f (1) = 1 is imposed as the initial condition. With regard to (B 1 ) L , the invariance can be easily seen by using e −q(h) b(z)eq (h) = e h(z) b(z) 8) and the fact that b(z) is a primary field with the dimension 2.
Now that the similarity transformation of (K ′ 1 ) L , (B 1 ) L and c(1) is established, we can transform the solution (2 . 12) to the original solution (2 . 11). An important point is that the generatorsq(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): Noting that U −1 f and eq (h) are given as an exponential of derivations, we find that the solution (2 . 12) is given as a transformation from (2 . 11): Let us consider the vacuum energy for Φ 0 around the trivial pure gauge solution. Using the transformation (2 . 31), the action for Φ 0 is given by where we have used (2 . 15) and the BPZ conjugation: . Since U f is generated by K n and the operators Q B and L n commute with each other, U f Q B U −1 f is equal to Q B . As a result, the vacuum energy for Φ 0 (K ′ , B, c) is equivalent to that for Ψ 0 (K, B, c) in the conventional theory.
Next, let us consider gauge invariant overlaps for Φ 0 (K ′ , B, c). The gauge invariant overlap for the open string field Ψ is defined as 19) O V (Ψ ) = I|V (i)|Ψ ,

Observables around the tachyon vacuum solution
Now let us consider observables for the classical solution (2 . 13) around the identity-based tachyon vacuum solution. In this vacuum, the modified K ′ Bc algebra and the classical solution are simplified as mentioned before, and Q ′ has vanishing cohomology. 2), 7) From Q ′ c = 0 in (2 . 8), c turns out to be an exact state with respect to the modified BRST operator Q ′ . Since Q ′ K ′ = 0, the solution (2 . 13) can be written as a modified BRST exact state: Therefore, we conclude that both of the vacuum energy and the gauge invariant overlaps are zero for the classical solution (2 . 13).
Here, we should note that the derivation of (2 . 37) requires careful consideration. In Ref. 7), the homotopy operator is given for the BRST operator Q ′ in the identity-based tachyon vacuum. In the case of the tachyon vacuum solution using the function (1 . 4) with a = −1/2, a corresponding homotopy operator isÂ = (b(1) + b(−1))/2. If it is used for the above exact form, such as χ = F (K ′ )Âc, 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 that the homotopy operatorÂ, such as {Â, Q ′ } = 1, is not unique because we can add to it a commutator [Q ′ , O], where O is an arbitrary operator with ghost number −2. Then, we have a possibility of providing a regularization procedure by adding such terms to the homotopy operator. In Ref. 17), several regularization methods are rigorously discussed.
In addition, we should notice that the divergence does not appear in the procedure used to analyze cohomology of Q ′ in Ref.
2). The string field c can be expanded in the Fock space for each L 0 -level and the highest level state in c is c 1 |0 : For the identity-based tachyon vacuum using the function (1 . 4) with a = −1/2, Q ′ has an oscillator expression: where R n is terms with the mode number n with respect to L 0 : * ) (2 . 40) * ) Here, we have expanded the conventional primary BRST current j B as j B (z) = ∞ n=−∞ Q n z −n−1 and therefore Q 0 = Q B in particular. The nilpotency of Q ′ leads to the anti-commutation relations, Then, Q ′ c is written by the Fock space state starting from a highest level state: (2 . 41) Therefore, we can solve the equation Q ′ c = 0 level by level and then c can be given by a modified 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 cohomology within the ghost number one sector. Thus, by solving the cohomology level by level, we can write c as a Q ′ exact state with a well-defined Fock space expression. As a result, the expression (2 . 37) can be well-defined with no divergence.

Comments on a simplified algebra
Here, we comment on the case that e h(z) for the identity-based solution has zeros on the unit circle but not at z = ±1. In Ref. 2), the identity-based solutions (1 . 1) with the function h l a (z) (l = 1, 2, 3, · · · ; a ≥ −1/2): were considered as a generalization of the function h a (z) (1 . 4), which is the case of l = 1 in the above. The solution corresponding to h l a (z) is pure gauge for a > −1/2 and we can apply the prescriptions in the previous subsections. In the case that a = −1/2, the corresponding solution is believed to represent the tachyon vacuum, where the BRST operator Q ′ around the solution has no cohomology, 2), 7) and we have It has zeros at z = ±1 when 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).
(2 . 47) Therefore, with t 1 such as e iθ 1 = tan(it 1 + π 4 ), or t 1 = arctanh(tan θ 1 2 ), c ′ defined in (2 . 44) can be expressed as c ′ = e it 1 K 1 c. Because K 1 = L 1 + L −1 is a derivation with respect to the star product, and noting [K 1 , (B 1 ) L ] = 0 and [K 1 , Q B ] = 0, we have and they form a kind of KBc ′ algebra. Furthermore, noting (2 . 6) and (2 . 7), we obtain a simplified algebra for a modified BRST operator Q ′ corresponding to the function h 2m −1/2 (z). Using the above, we can apply the prescription in §2.6 in a similar way. §3. Observables for identity-based solutions We consider direct calculation of observables for the identity-based tachyon vacuum solutions, by use of the method for identity-based marginal solution in Ref. 11).
We consider one parameter family of the identity-based solution, Ψ 0 (a). The parameter a deforms the function h(z) in the solution, and as the simplest case (1 . 4) it takes values a ≥ −1/2, the solution becomes the tachyon vacuum at a = −1/2 * ) and otherwise it is a trivial solution. In particular, we assume Ψ 0 (a = 0) = 0.
Here, we take Ψ a = Ψ 0 (a) + Φ 0 for a ≥ −1/2. We can easily find that Ψ a is a classical solution in the conventional theory, namely it satisfies QΨ a + Ψ 2 a = 0. Expanding the string field around Ψ a in the action, we have the kinetic operator Q Ψa : Q Ψa A = Q B A + Ψ a A − (−1) |A| AΨ a for all string field A. Q Ψa can be written as where Q ′ is the modified BRST operator in (1 . 3) and Q ′ Φ 0 represents the kinetic operator around the solution Φ 0 in the theory at the identity-based vacuum Ψ 0 (a). The important point is that we can construct a homotopy operator for Q ′ Φ 0 (= Q Ψa ) for a ≥ −1/2 by use of K ′ Bc algebra. * ) Differentiating the equation of motion, QΨ a + Ψ 2 a = 0, with respect to a, we find Since Q Ψa has vanishing cohomology, we have for some state Λ a . Integrating (3 . 3) from a = 0, we get where we have used the fact that in the case of a = 0, Ψ 0 (a = 0) = 0 and the K ′ Bc solution is the same as the conventional tachyon vacuum solution: Φ 0 (K ′ , B, c) = Ψ 0 (K, B, c).
From (3 . 4), we can calculate the gauge invariant overlap for the identity-based solution: where we have used the fact that the gauge invariant overlap is BRST invariant with respect to Q Ψa : O V (Q Ψa (· · · )) = 0. Noting that the formula (3 . 5) holds for a ≥ −1/2, by using the result of the gauge invariant overlap for Φ 0 in the previous section, the above is evaluated as where we have used the notation in Ref. 11). Thus, as expected for the identity-based solution, the gauge invariant overlap for a > −1/2 equals to that of trivial pure gauge solutions, and, in the case that a = −1/2, the gauge invariant overlap agrees with the result for the tachyon vacuum solution.
As emphasized in Ref. 12), the formula (3 . 4) is nothing but a gauge equivalence relation between Ψ 0 (a) + Φ 0 and Ψ 0 (K, B, c). In fact, given the relation (3 . 4), Ψ 0 (a) + Φ 0 can be written as (3 . 7) * ) In the case that Ψ 0 (K, B, c) is the Erler-Schnabl solution, 22) for example, we have Φ 0 (K ′ , B, c) =  We have constructed classical solutions Φ 0 in the theory expanded around the identitybased scalar solution Ψ 0 (1 . 1). We have taken advantage of the K ′ Bc algebra to calculate observables for the solution. In the case that Ψ 0 is trivial pure gauge, the observables for Φ 0 are equivalent to those for the corresponding solution Ψ 0 (K, B, c) in the original background.
In the case that Ψ 0 is the tachyon vacuum, they become equal to those for trivial 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 relation between Ψ 0 (a) + Φ 0 and Ψ 0 (K, B, c) which is regarded as a new expression for the identity-based solution. Thanks to this expression, we have calculated analytically observables for the identity-based scalar solution whether it corresponds to trivial pure gauge or tachyon vacuum.
Around the identity-based tachyon vacuum solution, the zeros of e h(z) on the unit circle play an crucial role in evaluating observables for Φ 0 . As seen in §2.7, there is no need for these zeros to be at z = ±1, which correspond to open string boundaries. We note that we can find similar results in study of homotopy operators for the BRST operator around the identity-based scalar solutions, 7), 23) in which homotopy operators exist only if the zeros are on the unit circle. Here, we should comment on another identity-based solution discussed in Ref. 3), in which e h(z) has higher order zeros than the function in this paper. However, similar 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 rather than the order.
For the simplest function h a (z) (1 . 4), the solution Φ 0 (K ′ , B, c) in the theory around Ψ 0 (a) depends on the parameter a. We find that for a > −1/2, Φ 0 can correspond to the tachyon vacuum but for a = −1/2, it becomes a trivial pure gauge configuration as stated in §2.6. This result is in accordance with the numerical analysis in Ref. 24), where it is observed that in the theory around Ψ 0 (a > − 1 2 ), we can construct a numerical solution whose energy density corresponds to the negative of the D-brane tension, while the solution continuously connects to trivially zero as a approaches −1/2. We find that the transition becomes sharp if the truncation level increases. Accordingly, Φ 0 can be identified as the numerical solution in Ref. 24) although they belong to different gauge sectors. Here, we should note that, for a = −1/2, we expect that there exists a solution whose energy density is the positive of the D-brane tension. Such a solution has been constructed numerically in the Siegel gauge in Refs. 9), 10). It should represent the perturbative vacuum where a D-brane exists. * ) Solving the equation (A . 1) under the initial condition f (1) = 1, the function f (z) is given by Since the function X(z) has singularity at z = ±i, v(z) becomes divergent and so the expression (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: where the first few coefficients are given by Using this expansion, we can construct a series solution for the differential equation (A . 1): where we find that A 0 = 0, λ = −e −h(i) and other A n is given by a recurrence formula: It can be easily seen that this series solution is convergent in a neighborhood of z = i. Since λ = −1 due to h(±i) = 0 for the identity-based solution, f (z) is a holomorphic function near z = i: Taking the limit z → i, we find that f (i) = iA 0 /A 0 = i. In addition, we have f ′ (i) = −1/A 0 = 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 solutions on the unit circle. Suppose that X(z) is expanded around the zero z 0 ( = ±i, |z 0 | = 1) as X(z) = x ′ 0 z − z 0 + x ′ 1 + · · · , (A . 9) g(z) behaves around z = z 0 as g(z) ∼ (z − z 0 ) 2ix ′ 0 × (· · · ), (A . 10) where the dots denote a regular function. Here it is noted that X(z) has poles on the unit circle, but the residue x ′ 0 is essentially unrestricted as opposed to the residue x 0 at z = ±i. Therefore, g(z) is not a regular function in general and so it is impossible to find a regular conformal transformation f (z) if e h(z) has zeros on the unit circle. Actually, we have seen an example for a singular map in §2. As mentioned in the introduction, the reality condition of Ψ 0 implies (h(z)) * = h(1/z * ) and so, for z = e iσ , h(z) is a real-valued function. Then, from (A . 11), we find that v(e iσ ) is real-valued. Consequently, from (A . 3), we find that |f (z)| = 1 for |z| = 1. Since v(e iσ ) and h(e iσ ) are real, the derivative is positive for |σ| ≤ π/2 and so φ(σ) is a monotonically increasing function from −π/2 to 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: The first is a Z 2 transformation derived from h(−1/z) = h(z), which is needed for the identity-based solution as mentioned in the introduction. The second transformation forms the 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). In fact, SO(2, C) has two real parameters and these correspond to integration constants for the complex first order differential equation (2 . 18). Then, since f (−1/z) is also a solution due to the first symmetry, we find the relation 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, 1). Consequently, since f (z) is holomorphic at z = i and f ′ (i) = 0, f (z) must satisfy the inversion formula (2 . 21).