Closed string vertex operators with various ghost number

We construct closed string vertex operators with various ghost numbers in addition to the conventional ones, using the Faddeev-Popov procedure for the gauge fixing of the conformal Killing group, from matter primary fields. We find that these operators give solutions to the descent equations in the framework of the BRST formalism. Similarly, we also construct solutions to the descent equations for the dilaton vertex operator with the Lorentz covariant form. Using the unintegrated vertex operator of the dilaton with the ghost number three, we obtain the correct result of the tadpole amplitude on the disk, including a non-zero contribution from a BRST exact term which comes from a conformal transformation.


Introduction
In the BRST formalism, physical states are defined as those that are annihilated by the BRST charge, and the physical observables are obtained by computing the cohomology of the BRST charge [1].In bosonic closed string theory, the BRST charge Q is given by the sum of the left and right-moving BRST charges, Q B and QB , and the physical states |phys are then defined as the states which satisfy [2] Q |phys = 0. (1.1) In addition to the BRST charge, there is a constraint known as the level-matching condition imposed on the closed string states: where L n and Ln are the left and right-moving components of the total Virasoro generators, respectively.In the BRST formalism, this is usually replaced by imposing the equivalent condition where b n and bn are the left and right-moving modes of anti-ghost fields, b(z) and b(z).Two conditions (1.1) and (1.3) provide semi-relative BRST cohomology, which is extensively used to investigate string amplitudes [3][4][5][6].Moreover, conventionally, covariant closed string field theory is formulated by the string field constrained by the level-matching condition.However, since the early days of closed string theory, the absolute BRST cohomology has been given by where c n and cn are the left and right-moving modes of ghost fields, c(z) and c(z), and |P i are transverse physical states in the matter sector [7].It should be noticed that the absolute cohomology includes states which do not satisfy the constraint (1.3).In addition, despite only the condition (1.1) being imposed, the cohomology automatically satisfies the level-matching condition (1.2).This result implies that the level-matching condition does not necessarily have to be imposed on physical states in the BRST formalism.Actually, some recent works investigate to construct closed string field theories without the level-matching condition [8,9].The absolute cohomology (1.4) includes states with ghost zero modes, and transverse states appear quadruply within the cohomology.Among the four sectors, cohomology proportional to c + 0 (= (c 0 + c0 )/2) is believed to be a BRST exact state.The state c + 0 |ψ , satisfying Qc + 0 |ψ = 0, can be formally expressed as c + 0 |ψ = Q((b 0 + b0 )/(L 0 + L0 ) c + 0 |ψ ).In the case being on-shell, this state becomes ill-defined due to L 0 + L0 being zero on |ψ .However, as discussed in [10], 1 using the continuity of L 0 + L0 , the state can be expressed as BRST exact through a limit approached from the off-shell region.Similarly, the avoidance of multiplicity is discussed in [7], using the relativistic spinless particle 1 See section 2.3 in [10].
as an example.Indeed, with these methods, it is easy to derive a BRST exact expression for certain on-shell states.
What is crucial here, as stated in [10], is that the argument using the continuity above cannot be applied to the cohomology proportional c − 0 (= c 0 − c0 )/2.This is due to the discrete spectrum of L 0 − L0 , which arises from constraints on the left and right momenta.Accordingly, only from the perspective of BRST cohomology, we have no reason to exclude states that include c − 0 -which does not conform to (1.3)-from the sets of physical states.
In this paper, we explore the possibility that the level-matching condition may not be as fundamental as previously thought in the BRST formalism and investigate closed string vertex operators with various ghost numbers, which are suggested to exist from the absolute BRST cohomology (1.4) and the state-operator correspondence.
The vertex operator for a closed string state is given by where V (z, z) is a matter primary operator of weight (1,1).The scattering amplitude is given by the Polyakov path integral with the insertions of these vertex operators integrated over a world-sheet.
Conventionally, the positions of some vertex operators are fixed to eliminate the gauge redundancy and the operators are multiplied by ghost fields:2 c(z)c(z)V (z, z). ( For fixed positions, dz ( dz ) in the integrated vertex operator is formally replaced by c(z) ( c(z) ) and then the operator increases the ghost number by two.Namely, differentials are related to ghost fields: It is noted that the operator (1.6) corresponds to the state of the first term in the absolute BRST cohomology (1.4).Following the correspondence of (1.7), it is possible to construct the closed string vertex operator with ghost number one.For instance, in the case that we fix the real part of z = x + iy and leave the integration along the imaginary axis, dx = (dz + dz)/2 is replaced by (c + c)/2 and then the vertex operator is given by3 dy (c(z) + c(z))V (z, z). (1.8) This operator is given also by acting an anti-ghost insertion associated with the y integration on the operator (1.6).The latter is a conventional procedure for a moduli integration and so this operator itself is not novel.Even if (1.7) is useful, it is impossible to reach the vertex operator with ghost number three, which corresponds to the state of the second and third terms in (1.4).In this paper, we will begin with a reconsideration of the gauge fixing of P SL(2, R) symmetry in a disk amplitude with a closed string insertion, and as a result, we will mainly propose that closed string vertex operators with various ghost numbers can be constructed by multiplying the operator to the operators (1.5), (1.6), (1.8) and so on.It is worth noting that the vertex operators with the factor (1.9) correspond to states that include c − 0 .Furthermore, we will explore these vertex operators with various ghost numbers, employing the perspective of differential forms.These vertex operators can be identified as operator-valued forms.Subsequently, we will establish that these forms can be related by descent equations with respect to the BRST transformation.Additionally, we will demonstrate that the supplementary terms incorporated into these forms during conformal transformation can be expressed as a sum of total derivative terms and BRST exact terms.Consequently, the amplitudes containing these forms exhibit conformal invariance, even though the factor (1.9) may initially appear to contradict this property.
Finally, we will apply the advanced technique of the descent equation for vertex operators to revisit the vertex operator associated with the dilaton.It is worth noting that the simple correspondence between (1.5) and (1.6) does not apply to the dilaton when its momentum is zero.Consequently, we will successfully construct vertex operators with various ghost numbers for the dilaton, including the momentum-zero dilaton.In particular, the vertex operator of dilaton with ghost number three represents a completely novel expression, which facilitates the calculation of the dilaton tadpole in the BRST formalism.
We will organize the paper as follows.In subsection 2.1, we provide a summary of our convention of vacuum, states, vertex operators and disk or sphere amplitudes.Specifically, we adopt the convention of treating the bra vacuum as the Hermitian conjugate of the ket vacuum for a sphere correlator, even though the often-used convention treats it as anti-Hermitian.In this subsection, we will elucidate how this convention naturally establishes a correspondence between an integrated and unintegrated vertex operator of closed strings.In the rest of section 2, we will examine closed string vertex operators in the BRST formalism using the Faddeev-Popov (FP) procedure to handle unconventional gauge fixing for the conformal Killing group (CKG) of a disk, namely, P SL(2, R).As a result, we will construct vertex operators with different ghost numbers, and we will observe that these operators include (1.9) as a factor.
In section 3, we treat these vertex operators as operator-valued forms on a world-sheet.We show that these forms satisfy the descent equations and examine their conformal transformation, thus ensuring the conformal invariance of amplitudes.From the descent equations, we will discover a new vertex operator with ghost number one, which is not present in the previous section.
In section 4, we will build the vertex operators of dilaton with various ghost numbers.An important distinction is that, unlike other vertices, the dilaton vertex with ghost number three and its ascendants incorporate the string coordinate X µ directly, rather than being derived from the derivative of X µ .The inclusion of X µ appears to be necessary when dealing with vertices associated with momentum zero.
In section 5, we use the dilaton vertex with a ghost number of three to compute the dilaton tadpole within the framework of the BRST formalism.Just like with other vertices, the BRST exact term is incorporated into the dilaton vertex during a conformal transformation.This additional BRST exact term doesn't affect the amplitudes for most vertex operators, but this isn't true for the dilaton vertex.This is because the correlation function involving BRST exact operators constructed using X µ may not necessarily equal zero, as discussed in [12].As a result, due to the non-zero contribution of this additional term, it is essential to establish the coordinate frame when defining the dilaton vertex initially.From the perspective of the state-operator correspondence, it is reasonable to consider that the dilaton vertex resides at the origin of a unit disk.Subsequently, by conformally mapping the unit disk to the upper-half plane, we evaluate the dilaton tadpole as a correlation function in the upper-half plane.By incorporating the contribution of the additional term, the resulting amplitude agrees with the established expression for the tadpole in [13].It is worth noting that the dilaton tadpole was originally derived in [14].Later, a subtle point was addressed in [13], emphasizing the necessity to include the integration of the world-sheet curvature in the calculation of the dilaton tadpole.In this paper, we compute the dilaton tadpole consistently within the BRST formalism by dealing with the dilaton as an unintegrated vertex operator.
Section 6 is devoted to concluding remarks.
2 Closed string vertex operators with various ghost numbers

Inner products and unintegrated vertex operators
Before proceeding further, let us examine the inner product of ground states.To simplify our analysis, we will assume that the space-time is flat Minkowski and all of the boundary conditions for the open string are Neumann.We adopt the definition of |0; k in [11] where the state is defined as the product of the matter ground state and the ghost ground state, with momentum k (see section 4.3 of [11]).We can identify this state with zero momentum as the following state in terms of the conformal field theory vacuum and ghost oscillators: closed string : where |0 refers to the SL(2, R) invariant vacuum state for open strings or the SL(2, C) invariant vacuum state for closed string. 4Using the normalization of the ground states from [11] (see (4.3.18a) and (4.3.18b) of [11]), which respect hermiticity, the inner product of these states can be expressed as5 open string : where δ 26 (0) is the delta function for momentum conservation.It is worth noting that, as stated in [11], the factor of i is required in the closed string case for hermiticity.When we factorize a correlation function into matter and ghost sector, we include i in the ghost correlation function.Expanding the ghost fields as the inner products (2.4) and (2.5) lead to the following correlation functions of the ghost sector: closed string : Once again, we observe the presence of the factor i in the closed string case, which is unconventional.However, we adopt this definition in order to clarify the correspondence between the one-form dz and the ghost field c(z).
When dealing with closed strings, according to this correspondence, the unintegrated vertex operator should be regarded as iccV (z, z).The expression, which has an unconventional factor of i, is naturally derived from the correspondence between the one-form and the ghost (1.7) as follows: where, in addition to the imaginary factor, we adopt the opposite sign compared to the closed string vertex operator in [11].This is feasible because the overall sign of ghost amplitudes are chosen to correspond to the positive FP determinant.We will verify that (2.8) and (2.9) lead to the conventional amplitude by examining the expectation value of closed string vertices.In the case of closed strings, the sphere amplitude can be related to the correlation function in the following way: where e −2λ represents a contribution related to the Euler characteristic in [11], C S 2 is a normalization constant for a sphere amplitude and the factor of i is inserted as discussed in [11].This factor is different from the previous i and is needed for the S-matrix calculations in Minkowski space-time.
The amplitude for three closed string vertices where the second and third factors correspond to ghost and matter sector, respectively.It should be noted that, in this case, the ghost sector is precisely equal to the FP determinant, despite the use of unconventional definitions.This is because the factor of i appears four times from the prefactor of the inner product and three closed string vertices, and the combined factor equals 1 due to i 4 = 1.Thus, this amplitude becomes equal to the conventional one.
Our convention for the inner product and the closed string vertex operator may seem unconventional compared to the ordinary ones.However, our convention is elegant and effective from the perspective of consistency when dealing with open-closed string amplitudes.Similar to the case of the sphere, the expectation value on the disk is related to the correlation function as where C D 2 is a normalization constant for a disk amplitude.For one closed and one open string amplitudes, the following ghost correlation function appears in the amplitude: where the coordinate z denotes the position of the closed string vertex on the upper half plane, while x represents the boundary coordinate of the open string vertex on the real axis.Using the correlation (2.7) and the ordinary doubling trick for ghosts, the correlation (2.13) is computed as i(z − z)|z − x| 2 and then it is real.Here, it should be noted that our convention for the closed string vertex operator includes a factor of i, which gives rise to a real-valued ghost correlation function.This is a contrast to the conventional definition of vertex operator, which does not have i and leads to an imaginary-valued correlation function.Here, it is noted that in the conventional case, an extra phase factor must be multiplied to the ghost correlation function corresponding to the FP determinant.However, one of the superior points of our convention is that it does not require such a complex phase.In the following section, we will demonstrate that our convention allows for a natural and straightforward construction of closed string vertex operators with various ghost numbers, without the need for complex factors that are required in the conventional approach.

A vertex operator with ghost number three
We begin with a disk amplitude with insertions of closed strings.We represent the disk as the upperhalf plane H + and so the CKG is P SL(2, R).Writing a closed string vertex operator explicitly, the amplitude is expressed as where g c is the closed string coupling constant assigned to the closed string vertex operator as in [11].
The abbreviation implies insertions of other vertex operators.The amplitude is defined by dividing the volume of P SL(2, R).The transformation θ ∈ P SL(2, R) is given by Here, dµ(θ) is the gauge invariant measure and, by using this measure, the gauge volume is formally expressed as If θ is parameterized by three positions x i (i = 1, 2, 3) on the real axis as in the conventional case, the measure is conventionally represented by and this expression fixes the normalization of the measure.In the standard way, the integration of (2.18) can be computed by using the infinitesimal transformation of P SL(2, R): (2.21) By following the standard technique, this FP determinant can be expressed as a correlation function of ghost fields inserted at the fixed positions of the closed and open string vertices.
Here, by integrating (2.18) and (2.21) with respect to x0 , we obtain the following equation:

22)
6 For z = x + iy (x, y ∈ R), the delta function is defined as (2.17) This equation corresponds to a kind of FP formula that applies when the position of only one closed string is fixed.This is made possible because the remaining gauge degree of freedom is compact, despite the fact that P SL(2, R) has three real degrees of freedom.Inserting the formula (2.22) into (2.14) and factorizing out the gauge volume (2.19), we can express the amplitude as This expression represents the amplitude with the position of only one closed string fixed, where the fixed vertex operator of the closed string has ghost number three.It can be observed that this vertex operator is obtained by multiplying the ghost operator (1.9) with the fixed closed string vertex, which includes the factor "i" as explained above.This fixed vertex operator corresponds to the state of a linear combination of the second and third terms of the absolute cohomology (1.4): Indeed, it is straightforward to establish the BRST and conformal invariance of the vertex operator.
It should be noted that the normalization of the vertex operator with ghost number three can be determined as 1/4π by the computation of a finite quantity.Based on the calculations of [14], which applied to the disk tadpole amplitude, obtaining the normalization factor involves a type of regularization due to the divergence of a functional determinant.However, our approach, which uses the FP technique, circumvents the need for regularization.
This vertex operator discussed above has ghost number three and could be potentially used to calculate the dilaton tadpole, as in [14].However, when the dilaton has zero momentum, a subtlety arises and a different approach is necessary.We will address this issue in later sections.

A vertex operator with ghost number two
Next, let us consider the gauge fixing condition in which we fix the position of one open string vertex and the real part of that of one closed string vertex: (2.25) Similarly to the previous case, it is possible to fix only two degrees of freedom of P SL(2, R) in this situation for the same reason.
To derive the FP formula corresponding to (2.25), we multiply (2.18) and (2.21) by |ẑ − ẑ * |: (2.26) Then by integrating it with respect to ŷ = Imẑ, we obtain the formula Applying the same technique of inserting this formula, we can obtain the amplitude involving ghost fields.However, it is important to note that the integration over ŷ introduces a discrete symmetry.
This residual symmetry is Z 2 obtained from combining the inversion of the real axis and an element of P SL(2, R).As a result, we can transform the position of the closed string vertex to another point with a different imaginary part.Therefore, the result obtained from the straightforward computation must be divided by two for the symmetry.The resulting amplitude under the gauge fixing (2.25) is expressed as where V c is a (1, 1) primary field for closed string, whereas V o is a primary field for open string with weight 1.The overall sign is given such that the FP determinant is positive.
We can easily confirm that the ghost number two operator produces desired results by computing an amplitude of one closed and two open string tachyons: where we define as x ij ≡ x i − x j .This result agrees with the correct amplitude, differing only by a sign factor.Thus, the operator appearing first in the amplitude (2.28) is the closed vertex operator with ghost number two.It should be noted that this operator includes an integration, while the conventional operator with ghost number two has a fixed position.Similar to the ghost number three vertex, we observe that this operator also consists of the conventional vertex (1.8) and the additional ghost operator (1.9).Furthermore, we will demonstrate that the operator is invariant under the BRST and conformal transformations, although the details of these invariances will be discussed in section 3.

Sphere amplitudes, gauge slices and generic paths
In this subsection, we will compute sphere amplitudes using the vertex operators constructed above, in order to verify their effectiveness on different topological surfaces.
Let us consider the amplitude for three closed string tachyons.First, we will evaluate using the vertex operator with ghost number three, and for the total ghost number, we will insert the vertex (1.8) with ghost number one as one of the three tachyons.The amplitude is given by the correlation function We can easily compute this correlation function and the result is This is the correct amplitude up to the sign factor.To obtain the full amplitude, we simply need to multiply this factor by the correlation function.Here, it is important to note that there are cases where the amplitude vanishes, namely when x 1 , x 2 > x 3 or x 1 , x 2 < x 3 .This seems to reflect that there are cases where our vertex operators do not fix the residual symmetry, such as P SL(2, C), correctly.By examining the disk amplitude as a simpler example, we illustrate the relationship between the choice of integration path and the gauge fixing of residual symmetry.We consider a disk amplitude of one closed and two open string tachyons.When using the closed string vertex (1.8) with the ghost number one, the amplitude is proportional to the following expression: (2.32) This correlation function is equivalent to the following integral, with the factor representing momentum conservation omitted: (2.33) This result implies that for the amplitude to be non-zero, the integration path must pass through the two points where open string tachyons are inserted.This can be understood by considering the following.Suppose that we fix first the position of the two open string vertices for gauge fixing of P SL(2, R).Then, it is seen that the residual symmetry allows us to move the position of the closed string vertex along the circle of Apollonius, which is generated by the points that have the same ratio of distances to the two open string vertex positions.Therefore, each of the Apollonius circles corresponds to a gauge orbit of the residual symmetry.Hence, the remaining integration path along the gauge slice must pass through the two open string vertices.
Here, it is worth noting that we can express (2.30) in terms of the integration along a generic path: where ∂D encloses a domain D. It can be shown by a simple computation that this expression is zero when the domain D includes both z 1 and z 2 , or neither z 1 nor z 2 .Otherwise, it provides the correct amplitude up to a sign, which depends on whether the domain D includes z 1 or z 2 .This generalization suggests that the vertex operator (1.8) is regarded as a one-form operator.Furthermore, this operator can be rewritten as where δ B O denotes the BRST transformation of the operator O.If we were to change the order of the z 3 -integration and δ B in this operator, it would become BRST exact, resulting in a zero amplitude.However, since this operator is considered as a distribution, the amplitude is given as a non-zero result.
In fact, we can explicitly observe that the correlation function with the insertion of δ B e ip 3 •X contains a delta function singularity, which leads to a non-zero result after the z 3 -integration.

Closed string vertices and descent equations
The equation (2.34) suggests that the vertex operator (1.8) corresponds to the one-form operator, Here, we adopt the notation that the subscript of ω denotes the rank of differential forms and the superscript indicates the ghost number of ω.Since V (z, z) is a (1, 1) primary field, and dzc(z) and dzc(z) have the weight (−1, −1), the form ω 1 1 is invariant under conformal transformations.
The BRST transformation of ω 1 1 is calculated as which leads to the BRST invariance of the integration of ω 1 1 when the surface term vanishes.Here, we find that the operator on the right-hand side of (3.2) is a conventional closed string vertex operator.Therefore, we introduce zero-and two-form operators that correspond to closed string vertex operators as follows: ω 2 0 and ω 0 2 correspond to conventional integrated and unintegrated vertex operators, respectively.
Similar to (3.1), these operators are also invariant under conformal transformations.Let d be the exterior derivative on forms, and assume that d and δ B commute.Then, we can find that these differential form operators satisfy the descent equations: In bosonic string theory, these descent equations were used to study the semi-relative condition for closed strings in [6].Additionally, in [15] and [12], a momentum zero dilaton state corresponds to a form that satisfies a part of the equations.In our case, (3.1), (3.3) and (3.4) are regarded as extended forms covering a wider range of quantum operators.
As seen in (2.23) and (2.28) in the previous subsections, we construct unconventional vertex operators with ghost number three and two.These correspond to the following differential form operators: These forms are given by multiplying the operator (1.9) to −ω 1 1 and ω 0 2 .Similarly, ω 2 1 is constructed using (1.9) and (3.3): and these satisfy the descent equations: These equations guarantee the BRST invariance of amplitudes that involve the corresponding vertex operators.
It is evident that ω 0 3 shares the same conformal invariance as ω 2 0 , ω 1 1 and ω 0 2 .However, in contrast to these operators, the operators ω 2 1 and ω 1 2 do not possess the conformal invariance.However, despite this lack of invariance, the amplitudes involving ω 2 1 or ω 1 2 remain invariant under conformal transformations.This is because these forms are transformed by the conformal transformation z ′ = f (z) as follows: ) where F is defined by The total derivative terms may vanish after integration and the BRST exact terms become zero in the amplitude.We can conclude that the forms for closed string vertex operators, which are constructed by the FP procedure, satisfy the descent equations.In addition, we have encountered ω 2 1 in the descent equations that is not derived from the FP procedure discussed in the previous section.Notably, ω 2 1 is a two-form that does not reduce the number of integrations, thus it adds to the ghost number without appearing to be directly related to gauge-fixing.Finally, we will check that ω 2 1 leads to a correct amplitude.As an example, we consider the amplitude of a closed string tachyon and two open string tachyons as follows, Since the two positions of open string vertices are fixed, an additional degree of freedom must be fixed for the gauge fixing of P SL(2, R).However, the current expression of the amplitude includes the integration over the position of the closed string vertex and does not fix any other degrees of freedom apart from the two positions.Consequently, this expression of amplitudes involves redundant integrations.Taking z = x + iy, the amplitude is calculated as This integration converges although only two positions are fixed and then the resulting amplitude is given by which coincides with the correct amplitude except for the sign factor, similar to (2.29).

Dilaton vertex operators
In this section, we examine dilaton vertex operators with various ghost numbers, which present a distinct situation in that they do not straightforwardly satisfy the conformal and BRST invariance as observed in the other vertices.

Dilaton two-forms with ghost number zero
Let us consider a dilaton vertex operator with ghost number zero: It is well-known that this is not a primary field for nonzero momentum.When the conformal mode of the world-sheet metric is taken into account, the dilaton vertex operator incorporates a term proportional to the scalar curvatures [16][17][18], which is essential for achieving conformal invariance.However, in accordance with the conventional BRST approach that does not incorporate the conformal mode, we initiate with the dilaton vertex operator (4.1).
To treat the dilaton vertex operator as a primary operator, independently of the conformal mode, various expressions can be considered.First, it is given by using a polarization tensor: where η µν is the metric tensor of the Minkowski space and pµ is an arbitrary vector satisfying p • p = 1 and p • p = 0.Although this expression represents a (1, 1) primary field, there is a subtlety concerning the momentum zero limit.Secondly, covariant expression was constructed by incorporating ghost and anti-ghost fields in [19]: which indeed is a (1, 1) primary field.When considering a two-form operator associated with these dilaton vertex operators, it might be necessary to consider adopting (4.3) in order to ensure both conformal and Lorentz invariance.However, it is easily found that both two-form operators constructed from (4.2) and (4.3) differ from the two-form of (4.1)only by a total derivative term: respectively.Consequently, we proceed with the following two-form operator constructed from (4.1): In fact, the total derivative term associated with Ω 2 0 provides a trivial contribution in solving the descent equations.Therefore, (4.5) alone is sufficient to address the descent equations and, in particular, construct a zero-form that follows (4.5).It is noted that in the limit of momentum zero, Ω 2 0 coincides with the two-form that has been investigated in [12].
It is worth noting that such a total derivative term becomes significant when integrating the twoform, as the resulting integration may capture the global structure of the world-sheet, similar to addressing the soft dilaton theorem [12,15].However, in this paper, we proceed with (4.5) as we do not delve into further integration of the two-form.

Descent equations
The form operators that satisfy the descent equations, including Ω 2 0 , can be readily obtained by iteratively applying the BRST transformation starting from Ω 2 0 .The resulting operators are given by These satisfy the following descent equations, Here, Ω 0 2 represents a well-known covariant expression of the dilaton vertex operator.It is worth noting that this expression remains unchanged regardless of whether we begin with the two-form given by the vertex operator (4.2) or (4.3).Let us explore the process of solving the descent equations of (3.8) for the dilaton vertex operators.First, we consider the BRST transformation of the zero-form which is obtained by multiplying Ω 0 2 with (1.9): Hence, it becomes necessary to introduce an additional operator to the zero-form in order to cancel out this term and achieve BRST invariance.One candidate for the additional operator is constructed using the vector pµ , which is used in the polarization tensor in (4.2): Actually, the BRST transformation of this operator is and so this operator contributes a cancellation of (4.9).However, this operator exhibits subtleties when considering zero momentum.Therefore, similar to the case of (4.2), we choose not to adopt it further.
Despite our attempt to solve the descent equations, starting with Ω 0 2 multiplied by (1.9), we were unable to identify the operators within the range of primary operators and their derivatives that cancel out the additional term (4.9).Hence, we encounter difficulties when attempting to directory multiply the operator (1.9) with Ω 2 0 , −Ω 1 1 or Ω 0 2 .
If we solely focus on solving the equations using forms with additional ghost numbers, we can identify the following forms as solutions for the descent equations: where Ω 2 0 , Ω 1 1 and Ω 0 2 are given by (4.5), (4.6) and (4.7).It is noted that Ω + 0 3 is exactly the BRST invariant vertex operator constructed in [19], which corresponds to the dilaton state proposed in [20].This operator corresponds to the state obtained by replacing c − 0 with c + 0 in (2.24), which is included in the absolute cohomology (1.4).For operators other than the dilaton, in general, we can solve the descent equations by multiplying ∂c + ∂c with ω 2 0 , −ω 1 1 and ω 0 2 as given in (3.3), (3.1) and (3.4).
Furthermore, we can solve the equations by using both ∂c + ∂c and ∂c − ∂c simultaneously.However, the cohomology with c + 0 should be a BRST exact state, as mentioned in the introduction.Indeed, probably in association with the BRST exactness, it can be observed that the amplitudes involving the operator ∂c + ∂c yield zero.Therefore, in this paper, we will not further explore the characteristics of the form in terms of ∂c + ∂c.

A dilaton vertex operator with ghost number three and ascendants
In this subsection, we will initiate the construction of dilaton vertex operators with ghost number three to address the descent equations with an additional ghost number.
As mentioned in the previous subsection, constructing a ghost number three primary operator for the dilaton, composed solely of conventional primary operators, appears to be an impossible task.So, we will explore such an operator within the extended space of operators including the string coordinates X µ .It is well-known that the operator X µ is not considered primary and so it is typically avoided when incorporating it into the framework of conformal field theory.However, X µ is originally incorporated as an operator in the first quantized string theory.Moreover, the inclusion of X µ offers intriguing insights into the dilaton theorem [12], and the BRST cohomology is well-defined when the zero-mode of X µ is included [21].Here, we will actively use X µ in order to construct the dilaton vertex operator.
From (4.9), we have only to construct an additional term ∆Ω 0 3 to cancel this anomalous term.To obtain such an operator, we first consider the following operators We can find the OPEs of C(z, z) with energy momentum tensors: and we can provide analogous OPEs for C(z, z).These OPEs result in the following BRST transformations of C and C: ) It is noted that these BRST transformations do not close within the set of prepared operators, and they additionally include p • Xe ip•X , ∂e ip•X and ∂e ip•X .Then, we extend the operators C and C by multiplying a series of p • X: where h(x) is a function defined by the series In addition, we define the operator for the function h(x) as The point is that these operators can be written as Moreover, we find the equation

p=p
= ĥ(p • X)e ip•X = e( ĥ), (4.22)where ĥ(x) = xh(x) − ixh ′ (x).Consequently, by applying h(−ip µ ∂/∂p µ ) to both sides of equations (4.16) and (4.17), and subsequently replacing p with p, we can derive the BRST transformation of C(h) and C(h) as shown below: c)e(24h + i ĥ).Finally, we conclude that, by using the function ∆Ω 0 3 is given by and this operator satisfies (4.11).As a result, the ghost number three vertex operator for dilaton is given by and, by construction, this is invariant under the BRST transformation.

Conformal transformations for the form of dilaton
First, we will confirm the conformal transformation property of the forms (4.5), (4.6) and (4.7).Under the conformal transformation z ′ = f (z), we find that the operator (4.1) is transformed as where the second and third terms in the right-hand side correspond to non-tensor terms.Therefore, the dilaton forms, Ω 2 0 , Ω 1 1 and Ω 0 2 , do not exhibit simple conformal invariance.
From (4.34), it follows that, under the conformal transformation z ′ = f (z), these forms are transformed as where F is defined by (3.11), resulting from the conformal map z ′ = f (z).Thus, the integration of Ω 2 0 satisfies conformal invariance similar to ω 2 1 and ω 1 2 in the previous section, and the extra terms vanish in amplitudes due to integration of forms and BRST invariance.For the forms (4.30), (4.33) and (4.32), we have to deal with the conformal transformation C(h) and C(h).From (4.14) and (4.15), we obtain the following OPEs:

.37)
Similar OPEs hold for C(h).Using these OPEs, we can deduce the transformation rules of C(h) and C(h) under the conformal mapping z ′ = f (z): Thus, the operators C(h) and C(h) are transformed as non-tensor fields.It is noted that, for arbitrary h, e(h) is transformed in the same way as a (0, 0) primary field, though it is not a well-behaved primary field due to X µ in e(h).Using (4.34), (4.38) and (4.39), we can find that, for the conformal mapping z ′ = f (z), the forms Ω 2 1 , Ω 1 2 and Ω 0 3 are transformed as ) where Φ 1 1 , Ψ 2 0 , Φ 0 2 and Ψ 0 2 are given by Hence, we have found that, under conformal transformation, all forms associated with dilaton are accompanied by a total derivative term and a BRST exact term.

Dilaton tadpole amplitude in the BRST formalism
Now that we have obtained the forms with various ghost numbers corresponding to closed string vertex operators, we can calculate various amplitudes by using these forms in the BRST formalism.In this section, even among them, we will focus on dilaton tadpole amplitude, which has been so far scarcely examined in the BRST formalism.The dilaton tadpole is computed from the amplitude with an insertion of a single dilaton vertex in an upper-half plane.Therefore, it is believed that this tadpole amplitude can be calculated using Ω 0 3 of (4.30), and the resulting amplitude will be examined to verify the well-known coefficient (26 + 2)/(26 − 2) discussed in [13].
The calculation of the first term is straightforward and yields: and the second term evaluates to zero.
To evaluate the third term, we need to consider the correlation functions of C(h), C(h) and e(h), which explicitly involve X µ operators.As discussed in [12], such a correlator should be evaluated using X µ (z, z) = −i ∂ ∂pµ exp(ip • X(z, z)) p=0 , and so, C(h) and C(h) should be replaced by Through "partial integration", this can be calculated as the product of the delta function and "a numerical factor".Therefore, the third term of (5.3) can be expressed as (5.8) It is important to emphasize that the numerical factor diverges and requires regularization.However, it should be noted that this divergence arises from the correlator of e(h), and in any case, we will continue with the calculation.
Regarding the fourth term, we need to evaluate the correlator of (5.9) It is easily found that the correlators of C(h), ∂e(h), ∂e ip•X and anti-holomorphic counterparts become zero based on the calculation from (5.6) and (5.7).Therefore, the fourth term is obtained by the correlator of e(h) + 1 8π e ip•X in (5.9).Since F is defined by (3.11) and f (z) in (5.9) is given by (5.2), we find that ∂F(z 0 , z0 ) = ∂F(z 0 , z0 ) = i Im z 0 .
(5.10) 11 According to [11], (5.4) space-time.We have explicitly shown that Ω g n is invariant up to δ B -exact and d-exact terms under the conformal transformations.It is characteristic that Ω 2 0 includes the ghost dilaton c∂ 2 c − c ∂2 c and Ω 3 0 is made of an infinite sum involving X • ∂Xe ip•X , X • ∂Xe ip•X .Using Ω 3 0 and its conformal transformation, we have evaluated the dilaton tadpole amplitude on the disk and obtained the correct result with the factor 26 + 2, which was calculated including a contribution from the curvature of the world-sheet in the previous literature.In our computation, we have found that there is a non-zero contribution from the δ B -exact term due to X µ .
As a closed string version of the calculations in [22][23][24], we can use the above ω g n to evaluate the closed string amplitudes with the mostly BRST exact operator.We will describe the details of this issue elsewhere.
15) for the real numbers a, b, c, d satisfying ad−bc = 1.Conventionally, since P SL(2, R) has three degrees of freedom, the positions of three open string vertex operators, or one closed and one open string vertex operators are fixed and multiplied by ghost fields in the gauge fixing procedure of P SL(2, R).Let us first consider the case that the positions of one closed and one open string vertex operators are fixed.The gauge fixing condition is (4.26), we can construct the supplementary term ∆Ω 0 3 by finding the function h which satisfies the equation: 24h(x) + i ĥ(x
Consequently, the correlators involving C(h) and C(h) in the third term are found to be zero.Similarly, the correlator for e(h) is evaluated asp µ 1 p µ 2 • • • p µ k ∂ ∂p µ 1 k