Superfield Approaches to a Model of Bosonic String: Curci-Ferrari-Type Restrictions

Abstract

Exploiting the theoretical potential of the modified Bonora-Tonin superfield approach (MBTSA) as well as the (anti-)chiral superfield approach (ACSA) to Becchi-Rouet-Stora-Tyutin (BRST) formalism, we derive the complete set of off-shell nilpotent (anti-)BRST symmetry transformations corresponding to the classical two- dimensional (2D) diffeomorphism symmetry transformations on the worldsheet (that is traced out by the motion of a model of bosonic string). Only the BRST symmetry transformations for this model have been discussed in the earlier literature. We derive the (anti-)BRST invariant Curci-Ferrari- (CF-) type restrictions (using MBTSA) which turn out to be the root cause behind the absolute anticommutativity of the above (anti-)BRST symmetry transformations. We capture the symmetry invariance of the (anti-)BRST invariant Lagrangian densities within the ambit of ACSA. The derivation of the proper anti-BRST transformations (corresponding to the already-known BRST transformations) and the (anti-)BRST invariant CF-type restrictions are the novel results in our present endeavor.

1. Introduction

Superfield approaches (see [1–8]) to Becchi-Rouet-Stora-Tyutin (BRST) formalism are geometrically elegant, mathematically rich, and physically very intuitive as they provide the geometrical basis for the off-shell nilpotency and absolute anticommutativity of the quantum (anti-)BRST symmetry transformations that are associated with a given classical local gauge symmetry transformation for a classically gauge invariant theory. In the above usual superfield approaches [1–8], only the -form () gauge theories have been considered which are characterized by the existence of the first-class constraints on them in the terminology of Dirac’s prescription for the classification scheme of constraints (see [9, 10]). It has been a challenging problem to incorporate the diffeomorphism invariant theories in the domain of the superfield approaches to BRST formalism. An attempt has been made by Delbourgo et al. (see [11]) in this direction where a diffeomorphism invariant gravitational theory has been considered. However, in our present endeavor, we shall not discuss anything connected with the superfield approach developed in [11] for the BRST analysis of our present two-dimensional (2D) diffeomorphism invariant theory.

A very successful application of the superfield approach [4–6] to BRST formalism (in the context of D-dimensional non-Abelian 1-form gauge theory) has been performed by Bonora and Tonin (BT). We have exploited the theoretical techniques and tricks of this approach in the context of BRST analysis of the higher -form () Abelian gauge theories [12]. It has been a very exciting problem to incorporate the diffeomorphism symmetry transformations within the framework of BT-superfield formalism. A breakthrough, in this direction, has been made by Bonora in a very recent paper [13] where the D-dimensional diffeomorphism invariant theory has been discussed within the ambit of BT-superfield approach [4–6]. We have christened this theoretical technique as the modified version of the BT-superfield approach (MBTSA) to BRST formalism [13] and applied its theoretical potential in the context of the 1D diffeomorphism (i.e., reparameterization) invariant model of a free spinning supersymmetric (SUSY) relativistic particle [14] and established that its Curci-Ferrari (CF) type of restriction as well as the gauge-fixing and Faddeev-Popov ghost terms are the same as for the other 1D diffeomorphism (i.e., reparameterization) invariant models of a free scalar and non-SUSY relativistic particle as well as a non-SUSY and nonrelativistic free particle (see [14] and the references therein).

In the applications of MBTSA [13], it turns out that we have to take into account the full super expansions of the superfields defined on the (D, 2)-dimensional supermanifold. In other words, we perform the superexpansion of the above superfields along all the possible Grassmannian directions of the (D, 2)-dimensional supermanifold on which a D-dimensional ordinary diffeomorphism invariant theory is generalized. The idea of horizontality condition (HC) enables us to derive the (anti-)BRST symmetry transformations for the scalars, vectors, tensors, etc. However, we invoke the Nakanishi-Lautrup-type auxiliary fields (with ) in the standard nilpotent (anti-)BRST symmetry transformations: of the (anti-)ghost fields in the case of the D-dimensional diffeomorphism invariant theory in an ad-hoc manner. This forces us to consider the (anti-)chiral superexpansions of the superfields (compare equation (26) below). At this juncture, it becomes essential for us to take into account the theoretical tricks and techniques of the (anti-)chiral superfield approach (ACSA) to BRST formalism (see [15] and the references therein) which has been developed by us.

The central theme of our present investigation is to apply the ideas of MBTSA and ACSA to BRST formalism in the realm of a 2D diffeomorphism invariant theory of a model of bosonic string and derive (i) all the (anti-)BRST symmetries of this theory in a consistent and clear fashion, and (ii) the CF-type restrictions which are responsible for the absolute anticommutativity of the (anti-)BRST symmetry transformations. We have also derived the BRST and anti-BRST invariant Lagrangian densities and captured their symmetry invariance(s) in the language of ACSA to BRST formalism. We would like to lay emphasis on the fact that the theoretical potential of MBTSA has been responsible for the derivation of (i) the (anti-)BRST symmetry transformations for the pure Lorentz scalars and (ii) the (anti-)BRST invariant CF-type restrictions. However, we have been able to derive all the proper (anti-)BRST transformations for all the other fields by using ACSA.

The following motivating factors have been at the heart of our present investigation. First, we have already used the beautiful blend of theoretical ideas behind MBTSA and ACSA in the cases of some 1D diffeomorphism (i.e., reparameterization) invariant theories of SUSY (i.e., spinning) relativistic particle, NSUSY (i.e., scalar) relativistic particle, and NSUSY and non-relativisticin physics system of a free particle for the discussion of BRST analysis. However, these models are also endowed with the gauge symmetry transformations which are a kind of a subset of the reparameterization symmetry transformations (under specific limits). To be precise, it has been shown (see [14] and the references therein) that the gauge symmetry transformations (generated by the first-class constraints) are equivalent to the reparameterization symmetry transformations if we use (i) the specific set of equations of motion and (ii) identify the transformation parameters of both these symmetries in a specific manner. Thus, it is a challenging problem for us to use the theoretical strength of MBTSA and ACSA in the context of a 2D diffeomorphism invariant theory which does not respect the gauge symmetry transformations as have been demonstrated in [14] for a 1D diffeomorphism invariant theory. We have discussed, in our present endeavor, a model of a bosonic string which has the 2D diffeomorphism symmetry invariance, but it does not respect a gauge symmetry transformation. Second, one of the sacrosanct aspects of BRST formalism is the existence of the quantum BRST and anti-BRST symmetries together for a given classical gauge/diffeomorphism symmetry transformation. For our present bosonic string, only the BRST symmetries are known in literature [16]. Thus, it is a challenge for us to derive the proper anti-BRST symmetry transformations corresponding to the above BRST symmetry transformations. We have accomplished this goal in our present endeavor. Finally, the hallmark of a BRST-quantized theory is the existence of the CF-type restrictions which provide the independent identity to the BRST and anti-BRST symmetries (and corresponding charges) at the quantum level. We have derived these restrictions, too.

The theoretical contents of our present endeavor are organized as follows. In Section 2, we concisely discuss the (anti-)BRST symmetry transformations for the gauge-fixed Lagrangian densities of the bosonic string theory. Section 3 is devoted to the derivation of the Curci-Ferrari- (CF-) type restrictions for our BRST invariant theory within the framework of MBTSA. In addition, we also derive the (anti-)BRST symmetry transformations for the target space coordinates and the determinant of the modified version of the inverse of the 2D metric tensor. Section 4 contains the derivation of the nilpotent (anti-)BRST symmetries for the other fields of our theory by exploiting the theoretical potential of ACSA. We capture the (anti-)BRST invariances of the Lagrangian densities using ACSA in Section 5. Finally, we make some concluding remarks in Section 6.

2. Preliminary: (Anti-)BRST Symmetries

We begin with the following (anti-)BRST invariant Lagrangian densities [] for the model of the bosonic string of our theory (see [17] for details) where the 2D diffeomorphism invariant classical action integral w.r.t. is as follows [16]:

In the above, we have taken the notation where is the evolution parameter (with ) and denotes the length of the bosonic string (with ). The modified version of the inverse of the 2D metric tensor is where is the inverse of the 2D metric tensor and . The coordinates (where ) correspond to the D-dimensional flat Minkowskian target space, and are the “time” and space directions on the worldsheet. The symbol denotes the string tension parameter, and is the Lagrange multiplier density which ensures that det so that we can have two degrees of freedom for the metric field tensor which, being symmetric, has only three degrees of freedom on a 2D flat spacetime manifold. In other words, we have (see [16] for details) the following decomposition of , namely,

The flat limit (i.e., ) can be obtained by the gauge-fixing conditions: . The latter choices imply that we have when we demand This input leads to for the choice where are the flat metric tensor and its inverse on the 2D Minkowskian spacetime manifold. In the derivation of the gauge-fixing and Faddeev-Popov ghost terms, we have taken the standard prescription of the BRST formalism (see [16]), namely, where the full set of nilpotent (anti-)BRST transformations are

Here, the fermionic [, etc.] (anti-)ghost fields are and the bosonic Nakanishi-Lautrup auxiliary fields are . From the above, we can derive the (anti-)BRST symmetry transformations for the component gauge fields and as follows:

It is interesting to note that these CF-type restrictions appear in the following simple cases of the proof of absolute anticommutativity property:

Thus, the off-shell nilpotent (anti-)BRST symmetry transformations (compare equations (6) and (5)) are the proper set of quantum symmetry transformations.

We end this section with the following remarks. First, the off-shell nilpotent (anti-)BRST symmetry transformations (6) and (5) correspond to the classical 2D diffeomorphism symmetry transformations: where is a physically well-defined function of on the 2D worldsheet such that it is finite at and but vanishes off as and . The infinitesimal version of these transformations are as follows: where (with ) are the 2D infinitesimal diffeomorphism transformation parameters. Second, according to the basic tenets of BRST formalism, the parameters have been replaced by the fermionic (anti-)ghost fields in the (anti-)BRST symmetry transformations (6) and (5). Third, it is crystal clear, from equation (9), that the (anti-)BRST symmetry transformations are absolutely anticommuting (i.e., ) in nature only on the submanifold of the quantum Hilbert space of fields where the CF-type restrictions are satisfied. Finally, we note that the target space coordinates and transform as pure Lorentz scalars (i.e., , ) under the infinitesimal and continuous diffeomorphism symmetry transformations: .

3. CF-Type Restrictions: MBTSA

According to the basic tenets of MBTSA to BRST formalism, first of all, we generalize the 2D infinitesimal diffeomorphism transformations to its counterpart onto the -dimensional supermanifold as follows (see [13, 18] for details): where the -dimensional supermanifold is parameterized by the superspace coordinates . Here, are the bosonic worldsheet coordinates and a pair of Grassmannian variables satisfies , . In equation (10), the fermionic (anti-)ghost fields are the ones that are present in the (anti-)BRST transformations (6) and (5). In view of the mappings established by Bonora and Tonin [4, 5], the coefficients of and in (10) have been taken to be the (anti-)ghost fields because, according to the standard BRST prescription, the classical infinitesimal diffeomorphism symmetry transformations have been promoted to the quantum level by the (anti-)BRST symmetry transformations: . The coefficients of in (10) (i.e., ) have to be determined from other consistency conditions of the BRST formalism which we elaborate as follows.

To derive the CF-type restrictions and the (anti-)BRST symmetry transformations , we generalize the target space ordinary coordinate fields onto the -dimensional supermanifold as follows: where are the superfields whose arguments incorporate the super diffeomorphism transformations (10) and, on the r.h.s., we have the secondary superfields which have the following superexpansions (as their arguments are transformations (10)), namely, where and the Taylor expansions have been taken around . Collecting the coefficients of , and , from the r.h.s. of the above equation, we obtain the following:

We note that the target space coordinate fields are the pure scalars with respect to the 2D worldsheet on which we have taken the diffeomorphism symmetry transformations . Thus, physically, it is evident that, ultimately, the restrictions on the (2, 2)-dimensional superfield is the following:

This is what has been called as the horizontality condition (HC) in [13, 18]. This HC (compare equation (14)) amounts to setting the coefficients of , and in expression (13) equal to zero. In other words, we have the following:

The last entry can be explicitly written by plugging in the values of and as follows:

Now, it is straightforward to check that we have the following:

As pointed out earlier, the coefficients of (i.e., ) in equation (10) and their presence in (17) can be computed by the requirements of the consistency conditions of BRST formalism.

One of the sacrosanct properties of a pure scalar field/superfield is the observation that it should not transform under any kind of internal, spacetime, supersymmetric, etc., transformations. As a consequence, the secondary superfields of the r.h.s. of (11) are as follows:

Similarly, the l.h.s. is . Substitutions of these equalities into (11) yield the following expressions in terms of , namely, in a view of the Bonora-Tonin (BT) mappings and which was established in the realm of the D-dimensional non-Abelian 1-form gauge theory [4, 5]. In fact, a close look at (19) demonstrates that this expansion is exactly like the BT-superfield approach to BRST formalism in the context of gauge theories. Thus, it is clear from (15) and (17) that we have obtained the following (in terms of the (anti-)BRST symmetry transformations of (6) and (5)), namely,

The absolute anticommutativity requirement (i.e., ) implies that the following equality is true, namely,

The explicit computations of and () yield where we have used and . In addition, we have taken and which are derived from the nilpotency requirements: and . The above equality (21) implies (from (22)) that we have which is nothing but the CF-type restrictions that have been obtained (compare equation (9)) from the requirement of the absolute anticommutativity property (i.e., ) of the (anti-)BRST symmetry transformations (6) and (5).

At this crucial juncture, we are in the position to determine the explicit expression for that is present in equations (10) and (17) by demanding the equality of each of the equations present in (22) with the expression for in (20). In other words, we find that

A close look at (24) implies that there are two ways to equate the l.h.s. (containing ) with the r.h.s. of the above equation, namely, which lead to the derivation of the CF-type restrictions (23). Thus, we conclude that the CF-type restrictions are hidden in the determination of of equation (10) by exploiting the absolute anticommutativity property (i.e., ) within the ambit of MBTSA to BRST formalism. Ultimately, we observe that the above logic can be repeated in the case of a pure scalar to derive the CF-type restrictions (23) and the (anti-)BRST transformations: and , too.

We wrap-up this section with the following remarks. First of all, we have taken the standard (anti-)BRST symmetry transformations which imply the following (in the terminology of the (anti-)chiral superfield approach (ACSA) to BRST formalism (see [15])), namely, where the superscripts and on the superfields (compare the l.h.s. of (26)) denote the chiral and antichiral versions of the full superexpansions and the subscripts and denote the fact that the coefficients of in the above expansions lead to the determination of BRST and anti-BRST symmetry transformations. In other words, we are sure about the nilpotent (anti-)BRST symmetry transformations in terms of the (anti-)chiral superfield expansions in equation (26). Second, it is the off-shell nilpotency requirements which lead to and . However, we have to obtain these transformations within the realm of the superfield approach. Furthermore, it is the requirement of the absolute anticommutativity properties which yield and . We have to obtain, however, these symmetry transformations too, by using the techniques of the superfield approach to BRST formalism which we accomplish in our next section. Third, we note that HC condition (14) has led to the following full superexpansion of the target space coordinate superfield, namely, where the superscript denotes the target space coordinate superfield that has been obtained after the application of HC which, ultimately, leads to (19). Here, the coefficients of and are the (anti-)BRST symmetry transformations that are listed in equations (6) and (5). Finally, we comment that an expansion like (27) can be also written for the derivation of the (anti-)BRST symmetry transformations for the scalar.

4. (Anti-)BRST Symmetries of Other Fields: ACSA

In this section, we exploit the theoretical strength of ACSA to BRST formalism (see [15] and the reference therein) to derive all the (anti-)BRST symmetry transformations (6) and (5) except such transformations for the target space coordinates and which have already been derived in the previous section by using MBTSA to BRST formalism [13, 18]. We are inspired to use, in our present section, ACSA to BRST formalism because of our observations in equation (26). First of all, we focus on the derivation of the BRST symmetry transformations (6) which have not been derived in the previous section. Thus, we wish to obtain . In this context, first of all, we generalize the ordinary 2D fields and onto a -dimensional antichiral super submanifold of the general-dimensional supermanifold as follows: where the 2D fields are fermionic secondary fields and is a bosonic secondary field due to the fermionic nature of the Grassmannian variable . The above -dimensional antichiral super submanifold is parameterized by where are the bosonic coordinates and is the fermionic Grassmannian variable. The superscript on the superfields denotes the antichiral superexpansions of the above antichiral superfields along the -direction of the above super submanifold.

The basic tenets of ACSA to BRST formalism require that the BRST invariant (i.e., quantum gauge invariant) quantities should be independent of the Grassmannian variables as the latter are only the mathematical artifacts that are useful in the context of theoretical techniques of SUSY theories. In this connection, we note that the following BRST (i.e., quantum gauge) invariant quantities are useful and important for us, namely,

The above invariant quantities are obtained by a close observation of the transformations (6) where an off-shell nilpotency property exists for the BRST-symmetry transformations. We focus on which implies the following restriction: where is the antichiral limit of the full superexpansion containing the nilpotent (anti-)BRST symmetries as the coefficients of and . In other words, we have the following:

Plugging in the appropriate superexpansions for from (28) as well as the superexpansion for from (31), we obtain the explicit expression for the secondary fields as . As a consequence, we have the following final expansion: where the subscript on the superfield (on the l.h.s.) denotes that the above antichiral superfield has been obtained after the application of the BRST invariant restrictions (30) and the coefficient of is nothing but the BRST symmetry transformation for the field which also encodes the following relationships: and . The latter establishes the connection between the nilpotency properties of and .

At this juncture, we now concentrate on the derivation of in the expansion of in equation (28). For this purpose, we note that . Following the basic principle of ACSA, the expressions in the square bracket have to be generalized onto the -dimensional antichiral super submanifold with the following BRST (i.e., quantum gauge) symmetry invariant restriction: where the expansions of and have been quoted in equations (32) and (28), respectively. Substitutions of these superexpansions into the l.h.s. and comparison with the r.h.s. of the restriction (33) lead to the following condition:

In other words, the restriction (33) implies that the BRST invariant quantity must be independent of . A careful and close look at the above equation leads to the following:

Substituting for we obtain the following from the above equation:

It is clear that the first term in the square bracket will be zero if we operate the derivative from outside. Thus, the final expression is as follows:

Integrating over and taking the physicality condition that all the fields must vanish off as and at , , we obtain the precise value of as follows:

Hence, we have the following final expansion for the superfield : which leads to the derivation of the BRST symmetry transformation as the coefficient of in the above equation implying, once again, that . This relationship establishes the connection between and translational generator along the -direction of the (2, 1)-dimensional antichiral super submanifold ,and it also demonstrates that (which is the connection between the nilpotency properties). It goes without saying that the subscript on the l.h.s. denotes that the superexpansion (39) has been obtained after the application of the BRST invariant restriction (33).

We now focus on the BRST invariance: . This observation can be generalized onto the (2, 1)-dimensional antichiral super submanifold with the following restriction on the antichiral superfields, namely,

The substitutions of expansions from (28) and (32) lead to the following equality:

In the above, the term can be written as . It is elementary to note that the second term will vanish off when we shall multiply by from the left (i.e., ). The substitution of the leftover term (i.e., ) into (41) leads to the following:

It is straightforward to note that satisfies the above equation very beautifully. Thus, we have, ultimately, the following expansion (compare equation (28)):

Hence, we have derived the BRST transformations as the coefficient of in the above superexpansion. It should be noted that the subscript on the superfield (compare l.h.s. of equation (43)) denotes that has been derived after the imposition of the BRST invariant restriction (40).

At this stage, we now wish to derive the BRST symmetry transformation using the theoretical strength of ACSA to BRST formalism. Towards this goal in mind, we have the following restriction on the antichiral superfields which have their superexpansions in (28) and (32), namely,

The above restriction has been obtained by a close look at the off-shell nilpotency property of the BRST symmetry transformations (6). This restriction on the antichiral superfields leads to the following condition on the basic and secondary fields: where we have used the superexpansions from (28) and (31). It is straightforward to note that the first five terms, above, lead to the following total derivative, namely, where we have used and . Adding and substracting , we obtain the following equation from (45):

Expanding the total derivative in the second entry of the above equation and rearranging these, we obtain the following interesting equation, namely,

Adding and subtracting , we finally obtain the following very nice-looking equation:

It should be noted that what we have added and subtracted in (48) is basically equal to zero on its own because we make the following observation:

In other words, the last entries in the second and third lines of equation (49) are zero on their own. We note that the symmetric indices in and antisymmetric indices in the square bracket are summed up to yield zero. It is straightforward now to point out that which satisfies equation (49). As a consequence, we have the following: where the coefficient of is nothing but the BRST symmetry transformation for that has been quoted in (6). The subscript on the l.h.s. of the above equation denotes that the antichiral superfield has been obtained after the application of the BRST invariant restriction on a specific combination of superfields (compare equation (44)).

We set out now to derive the anti-BRST symmetry transformations (5) by using ACSA to BRST formalism where first of all, we generalize the following basic and auxiliary fields of our theory onto a -dimensional chiral super submanifold: where are the fermionic and is the bosonic secondary fields that are to be determined in terms of the basic and auxiliary fields of the (anti-)BRST invariant Lagrangian densities (compare equation (1)). It is elementary to note that, in the limit , we retrieve the bosonic and auxiliary fields of . We point out that implies that we have where is the superfield that has been obtained after the restriction on the chiral superfield that is obtained in the generalization on the chiral super submanifold (which is parameterized by where characterize the 2D worldsheet and is the fermionic Grassmannian variable). The subscript denotes the chiral superfield which leads to the derivation of as the coefficient of in its expansion: . It should be further noted that we have not devoted time on the derivation of the (anti-)BRST symmetries that have already been derived and mentioned in Section 3 where the theoretical strength of MBTSA has been exploited.

A close and careful observation of the anti-BRST symmetry transformations (5) demonstrates that we have the following very useful and interesting combinations of fields: as the anti-BRST invariant quantities. The fundamental requirement of ACSA is that the generalizations of the quantities (present in the square bracket of (54)) onto a suitably chosen -dimensional chiral super submanifold should be independent of the Grassmannian variable . As a consequence, we have the following restrictions: where we have taken the superexpansions from (53) and is the chiral limit of the full expansion (compare (27)). In other words, we have the following: where the superscript denotes the chiral version of the full expansion of that has been obtained in the previous section (compare equation (27)).

We would like to lay emphasis on the fact that all the secondary fields and can be computed in an exactly similar manner as we have done in the case of determination of the BRST symmetry transformations for the superexpansions in equation (28). It turns out that, adopting this logic, we obtain the following:

Substitutions of the above secondary fields into the chiral superexpansions of equation (53), we obtain the following final superexpansions: where the subscript on the chiral superfields on the l.h.s. of the above equation (58) denotes that the above superfields have been obtained after the quantum gauge (i.e., anti-BRST) invariant restrictions on the chiral superfields (compare equation (55)) have been imposed. It can be readily checked that we have obtained the anti-BRST symmetry transformations as the coefficients of the chiral superexpansions in (58). It is nice to note that where the generic chiral superfield stands for the l.h.s. of (58) and the generic ordinary field.

We end this section with the following remarks. First, we have derived the (anti-)BRST symmetry transformations for the fields by exploiting the theoretical tricks of ACSA to BRST formalism. These fields are the ones for which the MBTSA has not been able to derive the (anti-)BRST symmetry transformations. Second, a careful and close observation of the theoretical contents of Sections 3 and 4 demonstrates that we have derived all the nilpotent (anti-)BRST symmetry transformations for our theory by exploiting the theoretical strength of MBTSA and ACSA. Finally, the (anti-)BRST symmetry transformations for the component fields and of (compare equation (3)) can be obtained from the exact expressions for and that have been quoted in (6) and (5). To be more transparent, we find the following antichiral superexpansions: where the coefficients of are nothing but the BRST symmetry transformations (compare equation (8)) on , and . In an exactly similar fashion, we can obtain the anti-BRST symmetry transformations on and from the following chiral superexpansions:

In the above, the coefficients of are nothing but the anti-BRST symmetry transformations for the component fields , and (compare equation (7)). We point out that the subscripts and in equations (59) and (60) have their straightforward meaning as we have established earlier. We lay emphasis on the fact that the superexpansions in (59) and (60) are very crucial and important as will be clear in the next section where we shall discuss the symmetry invariances.

5. Invariance of the Lagrangian Densities: ACSA

In this section, we capture the (anti-)BRST invariance of the Lagrangian densities (1) in terms of the (anti-)chiral superfields that have been obtained after the imposition of the (anti-)BRST invariant restrictions. In this connection, it is worth pointing out that we have already computed the BRST invariance of and anti-BRST invariance of in the ordinary space in our earlier work [17]. To be precise, the action integrals and remain invariant under the continuous, infinitesimal, and nilpotent transformations in (6) and (5). In this connection, first of all, we note that the following are true for the classical Lagrangian density , namely, and the total Lagrangian densities and transform as follows [17]:

The above observations demonstrate that and for the physical fields of the (anti-)BRST invariant theories which vanish off [16] at and due to Gauss’s divergence theorem. We mention, in passing, that and (where ) due to the (anti-)BRST transformations for in (61).

First of all, we capture the (anti-)BRST invariance of the action integral within the realm of ACSA. In this regard, we note the following (anti-)chiral generalizations of to its counterpart super-Lagrangians (i.e., ) on the -dimensional (anti-)chiral super submanifolds, namely, where the super-Lagrangian densities (on the l.h.s.) carry superscripts and to denote that these have been defined on the (2, 1)-dimensional (anti-)chiral super submanifolds of the (2, 2)-dimensional general supermanifold (that has been chosen for our discussion). The superfields with subscripts and as well as with superscripts , , , and have already been explained in our previous and present sections. Equation (61) can be captured in the superspace (where ACSA plays an important role). The mappings , lead to the following observations:

Thus, the (anti-)BRST symmetry invariances of have been expressed in the language of ACSA to BRST formalism. We have performed this exercise separately because, on its own, the original classical Lagrangian density transforms to the total derivatives (compare equation (61)) under the (anti-)BRST symmetry transformations.

We would like to express the symmetry transformations (62) and (63) in the realm of ACSA where the superexpansions in (26), (31), (32), (39), (43), (52), and (59) will be playing decisive roles for the BRST invariance (compare equation (62)). On the other hand, the superexpansions (26), (56), (58), and (60) will be very useful in capturing the anti-BRST invariance (compare equation (63)). With these inputs at our disposal, we set out to capture the BRST invariance in terms of and . Here, the latter is given in the language of the antichiral superfields that have been derived after the imposition of the BRST-invariant restrictions. These antichiral superfields might also be the limiting cases of the full superexpansions that have been derived in Section 3, namely, where we have taken the ordinary fields and because we know that due to the BRST invariance of ]. Ultimately, it turns out that we obtain the following due to the operation of on :

It is evident that the above equation captures the BRST invariance of the Lagrangian density in the superspace (as is clear from our observation on the r.h.s.).

We can repeat the same exercise for the anti-BRST invariance. For this purpose, first of all, we generalize to its counterpart chiral super-Lagrangian density on the (2, 1)-dimensional chiral super submanifold as where the ordinary fields and are present in the above super-Lagrangian density because which implies that . In other words, there is no chiral -dependence on the r.h.s. of the superexpansion of the superfield . The rest of the notations for the chiral superfields have already been explained earlier. At this juncture, in view of the mapping, we can capture the anti-BRST invariance (63) by applying a derivative on (68) which yields the following:

Hence, we have captured the anti-BRST symmetry invariance (63) in the language of ACSA to BRST formalism (as is evident from the r.h.s. of (69)).

We close this section with the following remark. We can capture the basic ideas behind the derivations of and which have been explained in equation (4). In view of the mappings , we can express the super (anti-)BRST invariant Lagrangian densities corresponding to the ordinary Lagrangian densities (compare equation (4)) as where all the symbols have been explained in our earlier discussion. It is crystal clear, from the above expression, that the (anti-)BRST invariance of the action integrals and can be captured in the terminology of ACSA to BRST formalism because and will be zero in the ordinary space. Furthermore, we note that and will always produce the total derivatives in the ordinary space thereby rendering the action integrals (i.e., and ) equal to zero (compare equation (70)). To be precise, the nilpotency property of the translational generators will ensure that and will be always the total derivatives in the ordinary space. Hence, we are able to capture the symmetry invariance(s) of the action integrals (corresponding to the Lagrangian densities and ) using ACSA.

6. Conclusions

In our present endeavor, we have exploited the theoretical potential of MBTSA and ACSA to derive all the (anti-)BRST symmetry transformations for the 2D diffeomorphism symmetry invariant model of a bosonic string theory. These symmetry transformations [] are proper because they are off-shell nilpotent of order two and absolutely anticommuting (i.e., ) in nature (compare equations (9), (6), and (5)). The latter property of the (anti-)BRST symmetry transformations [] is satisfied if and only if we invoke the sanctity of the CF-type restrictions (with ) which define a submanifold in the quantum Hilbert space of fields where the Nakanishi-Lautrup-type auxiliary fields as well as the (anti-)ghost fields are present algebraically in a specific manner (compare equation (23)). These restrictions are physical in some sense because they are (anti-)BRST symmetry invariant (compare equations (6) and (5)) on the above submanifold. Hence, their imposition on our BRST-quantized theory is logical.

By applying the theoretical strength of MBTSA, we have been able to derive, in one stroke, the (anti-)BRST symmetry transformations together for the Lorentz pure scalar fields (e.g., ) and the 2D version of the universal CF-type restrictions: . These 2D restrictions are the limiting case of the D-dimensional diffeomorphism invarant theory where the superfield approach (developed by us [13, 18]) leads to the existence of the D-dimensional CF-type restrictions (with ) where the fermionic (anti-)ghost fields correspond to the D-dimensional infinitesimal and continuous diffeomorphism symmetry transformations: . In these infinitesimal transformations, the parameters are the diffeomorphism transformation parameters. The symbols are nothing but the Nakanishi-Lautrup-type auxiliary fields in the D-dimensional theory. The existence of the D-dimensional CF-type restrictions is universal, and so far, their presence has been shown explicitly in the cases of 2D and 1D diffeomorphism invariant theories (see [14, 17] for details).

Within the ambit of MBTSA, it becomes evident that we have to take, at least, the help of the (anti-)chiral superfield expansions (compare equation (26)) so that we can obtain and for the D-dimensional diffeomorphism invariant theory (see [13, 18] for details) in addition to the validity of off-shell nilpotency property so that we can obtain: and . The above two inputs are essential for the completeness of MBTSA. Hence, we have exploited the theoretical potential of the ACSA to BRST formalism (see [15]) so that both the above inputs can be taken care of. As a consequence, it becomes important to blend together the ideas from the MBTSA and ACSA so that we can derive all the (anti-)BRST symmetry transformations for the all the fields of a diffeomorphism invariant theory along with the derivation of appropriate (anti-)BRST invariant CF-type restrictions. This is what we have precisely done in our present investigation. Our earlier works (see [14] and the references therein) on the 1D diffeomorphism invariant models of the relativistic and nonrelativistic particles (of SUSY and non-SUSY varieties) have also exploited the ideas behind MBTSA and ACSA together to obtain the 1D version of the universal D-dimensional CF-type restrictions that have been derived and thoroughly discussed in [13, 18].

In our earlier work [17] on our present bosonic string, we have computed the expressions for the BRST and anti-BRST charges in the flat space. In the paper by Kato and Ogawa [16], the nilpotency of the BRST charge has been proven to demonstrate that the quantum version of the theory is valid only when and . It will be a very nice future endeavor for us to take the expression for the anti-BRST charge and plug in the normal mode expansions of the fields (with creation and annihilation operators in it) so that the quantum version of it can be obtained. With appropriate boundary conditions on the target space coordinate fields and (anti-)ghost fields, it will be challenging to derive and from the requirement of the nilpotency of the anti-BRST charge in the flat limit. We are presently involved with this problem and our results/observations will be reported elsewhere.

As pointed out earlier, our present 2D diffeomorphism invariant theory is different from our earlier works on the 1D diffeomorphism (i.e., reparameterization) invariant theories (see [14] and the references therein) in the sense that the latter theoretical models have the gauge symmetry transformations, too, which are equivalent to the reparameterization (i.e., 1D diffeomorphism) symmetry transformations in the specific limits (see [14, 19] for details). It is worth emphasizing that the gauge symmetry transformations (generated by the first-class constraints) have been exploited for the BRST quantization in [19] in the cases of the 1D diffeomorphism (i.e., reparameterization) invariant models. The latter models are nothing but the non-SUSY scalar relativistic and SUSY spinning relativistic particles. We lay emphasis on the fact that the reparameterization symmetry transformations of these models have been left untouched in [19] as far as the BRST quantization scheme is concerned. We have taken this challenge in our earlier works (see [14] and the references therein) for the BRST quantization of these models.

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that there are no conflicts of interest.

Acknowledgments

Two of us (AT and AKR) would like to express their deep sense of gratefulness towards Banaras Hindu University (BHU) for its financial support through the BHU-fellowship. Prof. G. Rajasekaran is one of the very prominent mentors of our group, and all three of us would like to dedicate this work, very humbly and respectfully, to him on the occasion of his birth anniversary which was celebrated on February 2021.