# Nilpotent Symmetries of a Model of 2D Diffeomorphism Invariant Theory: BRST Approach

Authors: R. P. Malik
Journal: Advances in High Energy Physics
 (2022)

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## Abstract

Within the framework of the Becchi-Rouet-Stora-Tyutin (BRST) formalism, we discuss the full set of proper BRST and anti-BRST transformations for a 2D diffeomorphism invariant theory which is described by the Lagrangian density of a standard bosonic string. The above (anti-)BRST transformations are off-shell nilpotent and absolutely anticommuting. The latter property is valid on a submanifold of the space of quantum fields where the 2D version of the universal (anti-)BRST invariant Curci-Ferrari (CF) type of restrictions is satisfied. We derive the precise forms of the BRST and anti-BRST invariant Lagrangian densities as well as the exact expressions for the conserved (anti-)BRST and ghost charges. The lucid derivation of the proper anti-BRST symmetry transformations and the emergence of the CF-type restrictions are completely novel results for our present bosonic string which has already been discussed earlier in literature where only the BRST symmetry transformations have been pointed out. We briefly mention the derivation of the CF-type restrictions from the modified version of the Bonora-Tonin superfield approach, too.


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## Body

## 1. Introduction

One of the most exciting and captivating areas of research in theoretical high energy physics (THEP), over the last few decades, has been the subject of (super)strings and related extended objects (see, e.g., [1–4] for details). This is due to the fact that, in one stroke, these theories provide a possible scenario of unification of all the fundamental interactions of nature and a promising candidate for the precise theory of quantum gravity. The modern developments in the realm of (super)strings have influenced many other areas of research in THEP, e.g., noncommutative field theories, higher p-form (p=2,3,4,⋯) gauge theories, higher spin gauge theories, supersymmetric gauge theories and related mathematics, gauge-gravity duality, and AdS/CFT correspondence. The quantization of these (super)string theories have led us to imagine a higher dimensional view of the physical world we live in. It has been established that one cannot consistently quantize the dual-string theory [5] unless the space-time dimension is D=26 and the intercept (α0) of the leading Regge trajectory is α0=1. These results have been obtained and formally established from many different considerations like the requirement of the validity of proper Lorentz algebra, unitarity requirements of these string theories, nilpotency of the Becchi-Rouet-Stora-Tyutin (BRST) charge, etc. In this context, one of the earliest attempts to covariantly quantize a bosonic string theory, within the framework of BRST formalism, was undertaken by Kato and Ogawa [6] where the 2D diffeomorphism symmetry of this theory was exploited.In the above work [6], it is precisely the infinitesimal version of the 2D classical diffeomorphism symmetry invariance of the theory that has been primarily exploited to perform the BRST quantization where only the BRST symmetries have been discussed. However, there is no discussion about the anti-BRST symmetries and related Curci-Ferrari- (CF-) type restrictions which are the hallmarks of a properly BRST quantized theory. In this work [6], the inverse of the metric tensor has been taken in such a manner that the conformal anomaly does not spoil the BRST analysis. In fact, the inverse of the metric tensor has been decomposed such that it has three independent degrees of freedom to begin with. A Lagrange multiplier field density has been incorporated into the 2D Lagrangian density so that the equation of motion w.r.t. it puts a restriction on the determinant of the above metric tensor. The latter condition reduces the independent degrees of freedom of the metric tensor from three to two. The BRST charge has been calculated in the flat limit where the metric tensor becomes Minkowskian in nature (see, e.g., [6] for more details). The nilpotency requirement of this BRST charge leads to the derivation of D=26 and α0=1. One of the central theme of our present investigation is to focus on the existence of (i) the proper anti-BRST symmetries (corresponding to the BRST transformations taken in [6]) and (ii) the (anti-)BRST invariant CF-type of restrictions which are responsible for the absolute anticommutativity of the nilpotent (anti-)BRST symmetry transformations. We have taken a very modest step in this direction, in our present endeavor.We have performed the full BRST analysis of the above theory [6] in the sense that we have derived the proper anti-BRST symmetry transformations corresponding to the BRST symmetry transformations that have been taken into account in [6]. The BRST and anti-BRST symmetry transformations are found to be off-shell nilpotent and absolutely anticommuting in nature. The latter property has been shown to be true on a submanifold of the quantum Hilbert space of quantum fields where the CF-type restrictions (20) (see below) are satisfied. We observe that these restrictions are BRST as well as anti-BRST invariant thereby implying that these are physical restrictions (which can be imposed from outside on our present theory). We have derived, in our present endeavor, the BRST and anti-BRST invariant Lagrangian densities and have shown explicitly their BRST and anti-BRST invariance. The conserved charges of the theory have been computed in the flat limit where A0=A1=0,A2=1 (compare Equation (4)). In fact, the latter conditions imply that the metric tensor of the theory transforms as g~ab⟶ηab where ηab is the flat metric of the 2D Minkowski space (which is nothing but the 2D surface traced out by the propagation of the bosonic string). We have also established that the standard algebra between the ghost charge and BRST charge (as well as between the ghost charge and anti-BRST charge) is satisfied. We have commented, very briefly, on the nilpotency properties of the BRST and anti-BRST charges which are true at the quantum level only when D=26 and α0=1 provided we take into account the normal mode expansions of the fields (consistent with the appropriate boundary conditions) and substitute them in the computation and proof of nilpotency: QB2=1/2QB,QB=0 and Q¯B2=1/2Q¯B,Q¯B=0.The main motivating factors behind our present investigation are as follows. First, in the BRST description [6] of the present bosonic string, only the BRST transformations have been discussed corresponding to the infinitesimal diffeomorphism symmetry transformation of the theory. The nilpotent anti-BRST symmetry transformations have remained untouched in [6]. Thus, it is important for us to discuss the BRST as well as anti-BRST symmetry transformations together for the complete BRST analysis of our present theory. We have accomplished this goal in our present endeavor. Second, in the BRST description of Kato and Ogawa [6], the auxiliary fields have been modified/redefined in a very complicated fashion to simplify the theoretical analysis of the present theory. There are, however, no basic physical arguments to support such kinds of modifications/redefinitions. We have, in our present endeavor, not invoked any such kind of modifications/redefinitions as our analysis is very straightforward. Third, the hallmark of a quantum theory, discussed within the framework of BRST formalism, is the existence of the (non)trivial Curci-Ferrari- (CF-) type restrictions. We have derived such restrictions in our present endeavor which ensure the absolute anticommutativity of the (anti-)BRST symmetry transformations. Finally, our present work is important because, for this model, the recently developed superfield approach [7] would be very useful as our theory is diffeomorphism invariant. We hope that the application of this superfield formalism [7] would shed some new light on some specific aspects of our present theory (as far as the symmetries are concerned). In Appendix D, we have briefly discussed the applications of this superfield approach which has been christened by us as the modified version of the Bonora-Tonin superfield approach (MBTSA).Our present paper is organized as follows. To set up the notations and convention, we discuss very briefly the diffeomorphism symmetry as well as the corresponding BRST approach in Section2 which has been performed in [6]. Section 3 is devoted to the discussion of BRST and anti-BRST symmetries where we also point out the existence of the 2D version of the universal CF-type restrictions. We prove the (anti-)BRST invariance of this restriction, and we demonstrate the nilpotency as well as the absolute anticommutativity of the (anti-)BRST symmetry transformations. We derive the explicit form of the BRST as well as anti-BRST invariant Lagrangian densities in Section 4. The conserved charges, corresponding to the continuous internal symmetries of the theory, are derived in Section 5, in the flat limit. Finally, we make some concluding remarks on our present investigation in Section 6 and point out a few future directions for further investigations.In Appendices A, B, and C, we incorporate some of the algebraic expressions as well as equations that have been used in the main body of our text. In Appendix D, we concisely discuss the derivation of the CF-type restrictions by using the modified version of the Bonora-Tonin superfield approach (MBTSA) to BRST formalism.


## 2. Preliminary: Diffeomorphism and BRST Invariance

We begin with the Lagrangian density of a bosonic string theory as (see, e.g., [6] for details):
(1)L0=−12kg~ab∂aXμ∂bXμ+Edetg~+1,where g~ab=−ggab has two independent degrees of freedom because detg~=−1 due to the equation of motion w.r.t. the Lagrange multiplier field E which happens to be a scalar density (compare Equation (3)). Here, the 2D surface, traced out by the propagation of the bosonic string, is parameterized by ξa=ξ0,ξ1≡τ,σ where a=0,1, and component parameters (τ,σ) satisfy −∞<τ<+∞ and 0≤σ≤π. The string coordinates Xμξ (with μ=0,1,2,3,⋯,D−1) are in the D-dimensional flat Minkowskian space-time manifold and g~ab=−ggab is the metric tensor constructed with the determinant (g=detgab) and inverse (gab) of the metric tensor gab of the 2D parameter space. Under the infinitesimal diffeomorphism transformations ξa⟶ξa−εaξ, we have the following transformations δε on the relevant fields of our present bosonic string theory, namely
(2)δεXμ=εa∂aXμ,δεE=∂aεaE,δεdetg~=εa∂adetg~,(3)δεg~ab=∂mεmg~ab−∂mεag~mb−∂mεbg~am,where εaξ are the infinitesimal diffeomorphism transformation parameters. The above transformations leave the Lagrangian density (1) quasi-invariant (i.e., δεL0=∂aεaL0). This demonstrates that the action integral S=∫d2ξL0≡∫−∞+∞dτ∫0πdσL0 remains invariant under the diffeomorphism transformations (2) provided the boundary conditions εaξ=0 at σ=0 and σ=π are imposed on the diffeomorphism parameter εaξ. It will be noted that we differ from [6] by an overall sign factor in the diffeomorphism transformations (2) and BRST transformations (11) (see below) because we have chosen the infinitesimal diffeomorphism transformation ξa⟶ξa−εaξ, whereas the same transformation has been taken as ξa⟶ξa+εaξ in [6]. We choose the Latin indices a,b,c,⋯,l,m,n,⋯=0,1 to denote τ and σ directions on the 2D surface (traced out by the propagation of the bosonic string), and the Greek indices μ,ν,λ,⋯=0,1,2,⋯,D−1 stand for the spacetime directions of the D-dimensional flat Minkowskian space-time manifold corresponding to the target space. The above 2D surface is embedded in the D-dimensional Minkowskian flat target space (which turns out to be 26 at the quantum level). Throughout the whole body of our text, we denote the BRST and anti-BRST symmetry transformations by the symbols sB and s¯B, respectively. We adopt the convention of left-derivative w.r.t. the fermionic fields (Ca,C¯a, etc.) of our present theory. Consistent with this convention, the Noether conserved currents in Equations (25) and (26) are defined (see below).The original Lagrangian density,−1/2k−ggab∂aXμ∂bXμ (with k as the string tension parameter) is endowed with the local conformal invariance. However, this conformal invariance is broken by the conformal anomaly [8, 9] if we regularize the system in a gauge-invariant manner. We have avoided this problem by taking g~ab=−ggab as the metric tensor of our present theory [6] with three independent degrees of freedom to start with. The EoM w.r.t. E (i.e., detg~=−1) reduce the independent degrees of freedom of the above specifically defined metric tensor from three to two. For the BRST quantization of the Lagrangian density (1), we have to invoke the gauge-fixing conditions. This can be achieved if we take the following decomposition for the metric tensor g~ab (see, e.g., [6])
(4)g~ab=A1+A2A0A0A1−A2,and set the gauge-fixing conditions A0=A1=0 so that we obtain detg~=−A22=−1. This shows that, for the choice A2=1, we obtain the flatness condition g~ab⟶ηab with the signatures (+1, -1). By exploiting the standard techniques of the BRST formalism [10, 11], we obtain the gauge-fixing and Faddeev-Poppov ghost terms for the theory, in the language of the nilpotent (sB2=0) BRST transformations sB, as (see, e.g., [10, 11] for details)
(5)LGF+LFP=sB−iC¯0A0−iC¯1A1,where C¯0 and C¯1 are the antighost fields with ghost number (-1). It will be noted that the transformations sBC¯0=iB0 and sBC¯1=iB1 lead to the emergence of the Nakanishi-Lautrup-type auxiliary fields of the theory as B0 and B1 and the nilpotency requirements produce sBB0=sBB1=0. A close look at the transformations (2) and decomposition (3) leads to the following BRST symmetry transformations for the component gauge fields:
(6)sBA0=Ca∂aA0−∂0C1+∂1C0A1−∂0C1−∂1C0A2,(7)sBA1=Ca∂aA1−∂0C0−∂1C1A2−∂1C0+∂0C1A0,(8)sBA2=Ca∂aA2−∂0C0−∂1C1A1−∂1C0−∂0C1A0,where we have taken the replacement εa⟶Ca which implies that the infinitesimal diffeomorphism parameters (εa,a=0,1) have been replaced by the fermionic Ca2=0,CaCb+CbCa=0 ghost fields Ca. As a consequence of this replacement, we have the following BRST symmetry transformations vis-à-vis the transformations (2), namely
(9)sBXμ=Ca∂aXμ,sBE=∂aCaE,sBdetg~=εa∂adetg~,(10)sBCa=Cb∂bCa,sBC¯a=iBa,sBBa=0,(11)sBg~ab=∂mCmg~ab−∂mCag~mb−∂mCbg~am,where the transformation sBCa=Cb∂bCa has been derived from the requirement of the nilpotency condition (sB2Xμ=0). With the inputs from (6) and (11), we obtain the BRST invariant Lagrangian density (LB) from (5) and (1), modulo some total derivatives, as
(12)LB=L0+B0A0+B1A1+iCa∂aC¯0−C¯0∂aCa−C¯1∂0C1+∂1C0A0+iCa∂aC¯1−C¯1∂aCa−C¯0∂0C1+∂1C0A1−iC¯0∂0C1−∂1C0+C¯1∂0C0−∂1C1A2.The Lagrangian densityLB has been written, modulo some total derivatives, in such a manner that BRST transformations (6) could be implemented in a simple and straightforward manner. The above full Lagrangian density, under the flatness limit A0=A1=0,A2=1, reduces to the following Lagrangian density:
(13)LB⟶LB0=−12κηab∂aXμ∂bXμ+E1−A22+B0A0+B1A1−iC¯0∂0C1−∂1C0+C¯1∂0C0−∂1C1,which has been obtained in [6] after taking the help of the redefinitions of the auxiliary fields in a complicated fashion. In fact, these redefinitions are mathematical in nature, and there are no physical arguments to support the specific choices that have been made in [6] for the simplification of the Lagrangian density in the flat space. We have obtained (13) from (12) in a straightforward manner (without any redefinitions/modifications, etc.). We would like to point out that the flatness limit (i.e., A0=A1=0,A2=1) has been taken in all the terms of (12) except the gauge-fixing terms (i.e., B0A0+B1A1) and the Lagrange multiplier term (i.e., E1−A22) because the EoM w.r.t. B0,B1, and E imply the same thing (i.e., A0=A1=0,A2=1) in the straightforward fashion.


## 3. BRST and Anti-BRST Symmetries: Key Features

It can be checked, in a straightforward fashion, that the BRST symmetry transformations, quoted in (6) and (11), are nilpotent of order two (i.e., sB2=0). The proper anti-BRST symmetry transformations, corresponding to the BRST transformations (11), are
(14)s¯BXμ=C¯a∂aXμ,s¯BC¯a=C¯b∂bC¯a,s¯BCa=iB¯a,(15)s¯BE=∂aC¯aE,s¯Bdetg~=C¯a∂adetg~,s¯BB¯a=0,(16)s¯Bg~ab=∂mC¯mg~ab−∂mC¯ag~mb−∂mC¯bg~am,which are off-shell nilpotent (s¯B2=0) of order two. It will be noted that we have invoked a new Nakanishi-Lautrup type of auxiliary field B¯aξ in our theory. Thus, we observe that the symmetry transformations (16), (11), and (6) satisfy one (i.e., nilpotency) of the two sacrosanct properties (i.e., nilpotency and absolute anticommutativity) that have to be satisfied by any proper (anti-)BRST symmetry transformations. We further note that the last entry of (9) can be written in terms of A0,A1,A2 in the following form, namely
(17)s¯BA0=C¯a∂aA0−∂0C¯1+∂1C¯0A1−∂0C¯1−∂1C¯0A2,(18)s¯BA1=C¯a∂aA1−∂0C¯0−∂1C¯1A2−∂1C¯0+∂0C¯1A0,(19)s¯BA2=C¯a∂aA2−∂0C¯0−∂1C¯1A1−∂1C¯0−∂0C¯1A0.Thus, it is clear that the anti-BRST transformations forA0,A1,A2 are exactly the same as Equation (6) with the replacement Ca⟶C¯a.We dwell a bit now on the absolute anticommutativity property (i.e.,sB,s¯B=0) of the (anti-)BRST symmetry transformations (19), (16), (11), and (6). It turns out that the requirement of sB,s¯BXμ=0 lead to the existence of the following Curci-Ferrari- (CF-) type restrictions (which are primarily two in numbers), namely
(20)Ba+B¯a+iCb∂bC¯a+C¯b∂bCa=0,a,b=0,1.It turns out that the above conditions (20) have to be imposed to obtain the absolute anticommutativity (i.e., sB,s¯B=0) property when all the relevant fields of the whole theory are taken into account. For instance, it can be checked that the requirement of sB,s¯BE=0 also requires the validity of the CF-type restrictions (11). Furthermore, we obtain the following (anti-)BRST symmetry transformations on the Nakanishi-Lautrup auxiliary fields Baξ and B¯aξ due to the requirement of the absolute anticommutativity property (e.g., sB,s¯BCa=0 and sB,s¯BC¯a=0), namely
(21)sBB¯a=Cb∂bB¯a−B¯b∂bCa,s¯BBa=C¯b∂bBa−Bb∂bC¯a.Interestingly, the above transformations also satisfy the off-shell nilpotency property (i.e.,sB2=0,s¯B2=0) which is one of the key requirements of a proper set of (anti-)BRST symmetry transformations. Thus, we note that the (anti-)BRST symmetry transformations (21), (19), (16), (11), and (6) satisfy the off-shell nilpotency (sB2=s¯B2=0) and absolute anticommutativity (sB,s¯B=0) on a submanifold in the 2D Hilbert space of quantum fields where the CF-restrictions (20) are satisfied.We would enumerate here some of the subtle features associated with the CF-type restrictions (20) which are at the heart of the absolute anticommutativity property of our BRST and anti-BRST symmetry transformations. We note that these 2D restrictions on the auxiliary and (anti-)ghost fields are (anti-)BRST invariant quantity, namely
(22)s¯BBa+B¯a+iCb∂bC¯a+C¯b∂bCa=0,(23)sBBa+B¯a+iCb∂bC¯a+C¯b∂bCa=0.This demonstrates that the CF-type restrictions of our present theory are physical (in some sense) and the submanifold of the quantum fields in the Hilbert space defined by it is physically relevant. This demonstrates that our (anti-)BRST invariant theory is consistently defined on the submanifold where the CF-type restrictions (20) are always valid. In fact, on this submanifold alone, the BRST and anti-BRST symmetry transformations have their own identities as they are linearly independent of each other (due to their absolute anticommutativity). In the proof of (23), it is obvious that we have taken into account the (anti-)BRST transformations (21), (16), and (11) as well as the above specific submanifold.We end this section with the following remarks on the nilpotency properties (i.e.,sB2=0,s¯B2=0) associated with the BRST and anti-BRST symmetry transformations sB and s¯B. First of all, we note that sB2Xμ=0 leads to the derivation of sBCa=Cb∂bCa where sB2Ca=0 is also satisfied. In exactly similar fashion, we obtain s¯BC¯a=C¯b∂bC¯a (where s¯B2C¯a=0) from the requirement of nilpotency of the anti-BRST symmetry transformation on Xμ field (s¯B2Xμ=0). The proof of the nilpotency (i.e., sB2=0,s¯B2=0) of the transformations sBg~ab and s¯Bg~ab (compare Equations (11) and (16)) is algebraically more involved. We have collected some of the crucial expressions (as well as equations) in Appendix A which establish the nilpotency (sB2g~ab=0) of the BRST transformations when they act on the metric tensor g~ab. Ultimately, it turns out that sB2g~ab=0 and s¯B2g~ab=0 are indeed true. This proof, in turn, implies that the (anti-)BRST transformations (compare Equations (6) and (19)) of the component gauge fields (i.e., A0,A1,A2) of the metric tensor are automatically nilpotent (i.e., sB2=0,s¯B2=0) of order two (compare Equation (6)).


## 4. (Anti-)BRST Invariant Lagrangian Densities

We have already mentioned the BRST invariant Lagrangian densities (12) and (13) in the (non)flat limits. In our present section, we establish their BRST invariance. The analogue of the Lagrangian density (12) that remains invariant, under the anti-BRST symmetry transformations (16), (19), and (21), is as follows [10, 11]:
(24)LB¯=L0+s¯BiC0A0+iC1A1.Using the explicit anti-BRST symmetry transformations (16), (19), and (21), we obtain the following explicit form of the anti-BRST invariant Lagrangian density LB¯ as
(25)LB¯=L0−B¯0A0−B¯1A1+iC0∂aC¯a+∂aC0C¯a+C1∂0C¯1+∂1C¯0A0+iC1∂aC¯a+∂aC1C¯a+C0∂0C¯1+∂1C¯0A1+iC0∂0C¯1−∂1C¯0+C1∂0C¯0−∂1C¯1A2,where some total derivative terms have been dropped as they do not affect the dynamics of the theory. We shall take the flatness condition g~ab⟶ηab in the language of restrictions on the component gauge fields A0=A1=0,A2=1 for the full discussion of our theory within the framework of BRST formalism. In the flat limit (i.e., A0=A1=0,A2=1), the above Lagrangian density (i.e., (56)), in its full blaze of glory, is as follows:
(26)LB¯⟶LB¯0=−12κηab∂aXμ∂bXμ+E1−A22−B¯0A0−B¯1A1+iC0∂0C¯1−∂1C¯0+C1∂0C¯0−∂1C¯1,where the above limit has not been imposed on the gauge-fixing terms (−B¯0A0−B¯1A1) and the term E1−A22 with the Lagrange multiplier field. We shall be calculating the conserved charges of the theory from the Lagrangian densities (13) and (26) which are quoted in the flat limits (cf. Section 5 for details).To establish the explicit (anti-)BRST invariance of the Lagrangian densities (12), (13), (25), and (26), we have to apply the (anti-)BRST transformations on every term of the above Lagrangian densities. This exercise is algebraically more involved as one has to collect the terms containing A0,A1,A2,B0,B1, separately and independently. In Appendices B and C, we have collected these terms which appear due to the applications of sB and s¯B on the Lagrangian densities LB and LB¯, respectively. The explicit form of the BRST transformations on the BRST invariant Lagrangian density (7) (i.e., LB) is
(27)sBLB=∂aCaL0+B0A0+B1A1+iC¯1Ca∂0C1+∂1C0A0+iC¯0Cb∂bCaA0+iC¯0Ca∂0C1+∂1C0A1+iC¯1Cb∂bCaA1+iC¯0Ca∂0C1−∂1C0A2+iC¯1Ca∂0C0−∂1C1A2.In exactly similar fashion, the anti-BRST transformation acting on the anti-BRST invariant Lagrangian densityLB¯ produces the following explicit transformation:
(28)s¯BLB¯=∂aC¯aL0−B¯0A0−B¯1A1−iC1C¯a∂0C¯1+∂1C¯0A0−iC0C¯b∂bC¯aA0−iC0C¯a∂0C¯1+∂1C¯0A1−iC1C¯b∂bC¯aA1−iC0C¯a∂0C¯1−∂1C¯0A2−iC1C¯a∂0C¯0−∂1C¯1A2.A close and careful look at (27) and (28) shows that we can obtain (28) from (27) provided we make the replacements B0⟶B¯0,B1⟶B¯1,A0⟶−A0,A1⟶−A1,A2⟶−A2,C0↔C¯0,C1↔C¯1. Now it is obvious that, in the flat limit A0=A1=0,A2=1 of the full Lagrangian densities LB and LB¯, we obtain the following BRST and anti-BRST symmetry invariances for the Lagrangian densities LB0 and LB¯0, namely
(29)sBLB0=∂aCaL0+iC¯0Ca∂0C1−∂1C0+iC¯1Ca∂0C0−∂1C1,(30)s¯BLB¯0=∂aC¯aL0−iC0C¯a∂0C¯1−∂1C¯0−iC1C¯a∂0C¯0−∂1C¯1.The total derivatives in (27), (28), (29), and (30) establish that the (anti-)BRST transformations (21), (19), (16), (11), and (6) are the symmetries of the action integrals S=∫d2ξLB, S=∫d2ξLB¯, S=∫d2ξLB0, and S=∫d2ξLB¯0 provided we use the proper boundary conditions on the fields (and their derivatives) of the theory at σ=0 and σ=π [6].


## 5. Conserved Charges: Continuous Symmetries

The BRST chargeQB that has been computed in [6] is in the flat limit (A0=A1=0,A2=1) where the Lagrangian density LB0 (compare Equation (13)) plays a pivotal role. First of all, we note that the Lagrangian densities LB0 and LB¯0 (compare Equations (13) and (26)) respect the global ghost-scale symmetry transformations
(31)C0⟶eΩC0,C¯0⟶e−ΩC¯0,C1⟶eΩC1,C¯1⟶e−ΩC¯1,where Ω is a global scale transformation parameter. For the sake of brevity, we set Ω=1 so that the infinitesimal version (sg) of the above global scale symmetry transformations reduce to the following transformations on the (anti)ghost fields, namely
(32)sgC0=C0,sgC¯0=−C¯0,sgC1=C1,sgC¯1=−C¯1.Here, the subscriptg denotes the infinitesimal ghost-scale transformations (where Ω=1). The ghost charges, computed from LB0 and LB¯0, are as follows:
(33)Qg=∫0πdσJg0≡−i∫0πdσC¯0C1−C¯1C0,(34)Q¯g=∫0πdσJ¯g0≡−i∫0πdσC¯1C0−C¯0C1,where Jg0 and J¯g0 are the zeroth component of the Noether conserved currents (corresponding to the infinitesimal ghost transformations (32)) that have been derived from LB0 and LB¯0, respectively. However, the above charges are not independent of each other. Rather, they differ by a sign factor only (i.e., Qg=−Q¯g). Using the following Euler-Lagrange equations of motion that emerge out from LB0, namely
(35)∎Xμ=0,A0=A1=B0=B1=0,A2=1,E=0,∂0C¯0+∂1C¯1=0,(36)∂0C0−∂1C1=0,∂0C1−∂1C0=0,∂0C¯1+∂1C¯0=0,we observe that Q̇g=i∫0π∂/∂σC¯1C0−C¯0C1=0 due to the boundary conditions. This shows that the ghost charge is conserved (i.e., Q̇g=0).We now concentrate on the derivation of the BRST chargeQB and anti-BRST charge Q¯B¯ from the Lagrangian densities LB0 and LB¯0, respectively. Taking into account the basic concepts behind the Noether theorem, we note that QB=∫0πdσJB0,Q¯B=∫0πdσJ¯B0 where JB0 and J¯B0 are the zeroth components of the Noether conserved currents (corresponding to the BRST and anti-BRST symmetry transformations) computed from the Lagrangian densities LB0 and LB¯0, respectively. The explicit expressions for these currents, derived from the above Lagrangian densities, are
(37)JB0=sBXμ∂LB0∂∂0Xμ+sBC0∂LB0∂∂0C0+sBC1∂LB0∂∂0C1+sBC¯0∂LB0∂∂0C¯0+sBC¯1∂LB0∂∂0C¯1−X0,J¯B0=s¯BXμ∂LB¯0∂∂0Xμ+s¯BC0∂LB¯0∂∂0C0+s¯BC1∂LB¯0∂∂0C1+s¯BC¯0∂LB¯0∂∂0C¯0+s¯BC¯1∂LB¯0∂∂0C¯1−Y0,where the explicit expressions for X0 and Y0 are
(38)X0=C0L0+iC¯0C0∂0C1−∂1C0+iC¯1C0∂0C0−∂1C1,Y0=C¯0L0−iC0C¯0∂0C¯1−∂1C¯0−iC1C¯0∂0C¯0−∂1C¯1.The above expressions are derived from Equations (29) and (30) which are nothing but the zeroth components of the expressions that have been quoted in the square brackets. Finally, we obtain the following expressions for the conserved BRST and anti-BRST charges (QB and Q¯B) from the Lagrangian densities LB0 and LB¯0, namely
(39)QB=−∫0πdσC02κ∂0Xμ∂0Xμ+∂1Xμ∂1Xμ+C12κ∂0Xμ∂1Xμ+∂1Xμ∂0Xμ+iC¯0Ca∂aC1+iC¯1Ca∂aC0,(40)Q¯B=−∫0πdσC¯02κ∂0Xμ∂0Xμ+∂1Xμ∂1Xμ+C¯12κ∂0Xμ∂1Xμ+∂1Xμ∂0Xμ+iC0C¯a∂aC¯1+iC1C¯a∂aC¯0,where we have used the Euler-Lagrange (EL) equations of motion (EoM) (36) derived from the Lagrangian density LB0 and the following EL-EoM that emerge from the Lagrangian density LB¯0, namely
(41)∎Xμ=0,A0=A1=A2−1=B¯0=B¯1=E=0,∂0C¯0−∂1C¯1=0,(42)∂0C¯1−∂1C¯0=0,∂0C1+∂1C¯0=0,∂0C¯0+∂1C¯1=0.In fact, a close and careful look at the EL-EoM (36) and (42) establishes the fact that X0=Y0=0 on the on-shell (because we substitute the EL-EoM into them).The above chargesQB and Q¯B are conserved. This can be checked by exploiting the strength of the EL-EoM (36) and (42) while we take into account the direct “time” derivative of the above charges, namely
(43)Q̇B=−∫0πdσ∂∂σC02κ∂0Xμ∂0Xμ+∂1Xμ∂1Xμ+C12κ∂0Xμ∂1Xμ+∂1Xμ∂0Xμ+iC¯0Ca∂aC0+iC¯1Ca∂aC1,Q¯̇B=−∫0πdσ∂∂σC¯02κ∂0Xμ∂0Xμ+∂1Xμ∂1Xμ+C¯12κ∂0Xμ∂1Xμ+∂1Xμ∂0Xμ+iC0C¯a∂aC¯0+iC1C¯a∂aC¯1.The above expressions demonstrate that the BRST and anti-BRST charges are conserved when we use the boundary conditions atσ=0 and σ=π on the appropriate fields and their derivatives (see, e.g., [6] for details). Thus, we have noted that there are three conserved charges (which correspond to three continuous symmetries that are present) in the theory. One can check, in a straightforward manner, that the ghost charge obeys the standard algebra with the BRST and anti-BRST charges. This can be checked in a simple manner by computing the left hand side of the following from (32), (34), (39), and (40), namely
(44)sgQg=−iQg,Qg=0,sgQB=−iQB,Qg=QB,sgQ¯B=−iQg,Q¯B=−Q¯B,which demonstrates that we have iQg,QB=+QB and iQg,Q¯B=−Q¯B. However, the proof of nilpotency of the BRST and anti-BRST charges requires very careful computations at the quantum level where the normal mode expansions of the fields of our theory play very important roles. In the paper by Kato and Ogawa [6], this exercise has been performed, and it turns out that the nilpotency of the BRST charge is true only when D=26 and α0=1. It is obvious that we shall get the same result if we check the nilpotency of the anti-BRST charge at the quantum level with the proper boundary conditions.


## 6. Conclusions

In our present investigation, we have been able to derive the proper anti-BRST symmetry transformations corresponding to the BRST transformations (that have been shown to be present for the model of 2D diffeomorphism invariant bosonic string theory [6]). The BRST and anti-BRST symmetry transformations are proved to be off-shell nilpotent of order two. However, these symmetries are found to be absolutely anticommuting only on a submanifold of the Hilbert space of quantum fields that is characterized by the 2D field Equation (20). These latter equations are nothing but the CF-type restrictions which are the hallmark of the quantum diffeomorphism/gauge-invariant theories when these theories are discussed within the framework of BRST formalism. In fact, it is the existence of the CF-type restrictions that primarily imply that the BRST and anti-BRST symmetries (and the corresponding conserved charges) have their own identities. In the language of mathematics, they are linearly independent of each other (on a submanifold of the quantum Hilbert space of fields) that is defined by the CF-type restrictions (20).We have derived, in our present endeavor, the explicit and separate forms of BRST and anti-BRST invariant Lagrangian densities, and we have demonstrated clearly their transformation properties under the BRST and anti-BRST symmetry transformations. Using the Noether theorem, we have computed the conserved BRST, anti-BRST, and ghost charges of the theory in the flat limit. In fact, in the latter limit, the BRST charge has also been derived by Kato and Ogawa [6]. We have shown that the standard algebra is obeyed between the ghost charge and BRST charge (as well as the ghost charge and anti-BRST charge). The nilpotency (QB2=0,Q¯B2=0) of the BRST (QB) and anti-BRST (Q¯B) charges has not been derived in our present investigation as this requires the normal mode expansion of the fields and their substitution in the expressions for QB and Q¯B. In fact, the requirement of the nilpotency of the BRST charge has led to the derivation of D=26 and α0=1 where D is the dimensionality of the target space-time manifold and α0 is the intercept in the Regge trajectory that is generated due to the concept of strings (see, e.g., [6] for details).We would like to comment a bit on the boundary conditions that are to be imposed on the fields (and the derivatives on them) in our present theory when we demand the BRST as well as anti-BRST invariance of the Lagrangian densities (12) and (25). For the BRST invariance of the theory, the boundary conditions that have been obtained in the work by Kato and Ogawa [6] are ∂1Xμ=0,C¯0=0,C1=0 at σ=0 and σ=π. The BRST invariance of the boundary condition C1=0 (at σ=0 and σ=π) leads to the further boundary condition as ∂0C1=0 at σ=0 and σ=π. The anti-BRST invariance, in exactly similar manner, would lead to the boundary conditions ∂1Xμ=0,C0=0,C¯1=0 at σ=0 and σ=π. The anti-BRST invariance of the condition C¯1=0 at σ=0 and σ=π implies that ∂0C¯1=0 (at σ=0 and σ=π). Thus, the normal mode expansions of the fields Xμτ,σ,C0τ,σ,C1τ,σ,C¯0τ,σ,C¯1τ,σ can be found in the same manner as has been obtained in the work by Kato and Ogawa [6]. We have to be just careful that for the anti-BRST invariance; the mode expansions in the ghost sector should be such that the expansions are exchanged, namely Ca⟷C¯a. The requirements of the nilpotency of QB and Q¯B would obviously produce the results D=26 and α0=1.We would like to mention that the BRST and anti-BRST invariant Lagrangian densities (12) and (25) have been derived in a straightforward manner by utilizing the gauge-fixing A0=A1=0 and the (anti-)ghost fields (compare Equations (5) and (24)). However, if we compute the Lagrangian densities in the Curci-Ferrari gauge [12, 13], that would give due respect to the CF-type conditions that have been derived in (20). We wish to devote time on the computation of the coupled Lagrangian densities (like the 4D non-Abelian gauge theory [10–13]) which produce the CF-type condition as the equations of motion. Furthermore, the coupled Lagrangian densities should respect both the BRST and anti-BRST symmetry transformations on the submanifold of the quantum Hilbert space of fields where the specific quantum fields obey the CF-type restrictions (20). At present, we are working in this direction, and our results would be reported elsewhere.In a very recent work [7], the superfield approach to derive the proper (anti-)BRST symmetry transformations for any general D-dimensional diffeomorphism invariant theory has been developed (corresponding to its diffeomorphism symmetry invariance). It would be a very nice future endeavor for us to apply the theoretical arsenals of this superfield formalism [7] to our present bosonic string model which is also a diffeomorphism invariant theory. In fact, we hope that this superfield formalism would be able to shed more light on the geometrical origin and interpretation of the (anti-)BRST symmetries and the CF-type restrictions (20) which we have obtained for our present theory. In our earlier works on the Abelian 2-form and 3-form gauge theories [14, 15], we have established the geometrical origins and interpretations for the CF-type restrictions and their intimate connections with the geometrical objects called gerbes. It would be a challenging future endeavor for us to establish the same type of connections for our present 2D diffeomorphism invariant theory where the nontrivial CF-type restrictions exist. We are presently involved with this problem, and we plan to report about our progress in our future publication(s) [16].It is gratifying to state that we have already exploited the beauty and strength of MBTSA in the cases of 1D diffeomorphism (i.e., reparametrization) invariant interesting models of nonrelativistic free particle [17], scalar relativistic particle [18], and spinning (i.e., SUSY) relativistic particle [19] and established the universality of the 1D CF-type restriction B+B¯+iC¯Ċ−C¯̇C=0 in all the above non-SUSY and SUSY systems of interest and obtained the proper (anti-)BRST symmetry transformations corresponding to the classical 1D diffeomorphism (i.e., reparameterization) symmetry transformation. Here the (anti)ghost variables C¯C are the generalization of the infinitesimal reparametrization symmetry transformation parameter ετ in the transformation τ⟶τ−ετ where τ is an evolution parameter (see, e.g., [17–19] for details). In our present 2D diffeomorphism invariant bosonic string theory, we have found the CF-type restrictions as Ba+B¯a+iC¯b∂bCa+Cb∂bC¯a=0 (with a,b,=0,1) which are, once again, the limiting case of the MBTSA to D-dimensional diffeomorphism invariant theory where it has been shown [7, 20] that the CF-type restrictions Bμ+B¯μ+iC¯ρ∂ρCμ+Cρ∂ρC¯μ=0 (with μ,ν,ρ⋯=0,1,2,⋯D−1) are universal for the SUSY and non-SUSY systems in any arbitrary dimension of space-time where the (anti)ghost fields C¯μCμ are the quantum generalizations of the infinitesimal D-dimensional diffeomorphism transformation parameters εμx in the transformations xμ⟶xμ−εμx. In the above discussions, all the B¯B fields, with appropriate index, are the Nakanishi-Lautrup-type auxiliary fields.



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