Path-integral quantization of tensionless (super) string

: In this work, we study the tensionless (super)string in the formalism of path-integral quantization. We introduce BMS bc and βγ ghosts intrinsically by accounting for the Faddeev-Popov determinants appeared in ﬁxing the gauges. We then do canonical quantization and obtain the critical dimensions for diﬀerent tensionless strings. We ﬁnd that among four kinds of tensionless superstrings, the N = 2 homogeneous and inhomogeneous doublet tensionless superstrings have the same critical dimension as the usual superstrings. Taking the BMS bc and βγ ghosts as new types of BMS free ﬁeld theories, we ﬁnd that their enhanced underlying symmetries are generated by BMS-Kac-Moody algebras, with the Kac-Moody subalgebras being built from a three-dimensional non-abelian and non-semi-simple Lie algebra.


Introduction
It is well-known that the string scattering amplitudes behave in a particularly simple way in the high energy limit α → ∞ [1,2]. There are infinitely many linear relations among these scattering amplitudes, which indicate the existence of a higher symmetry structure in this tensionless limit [3]. In a modern perspective, the study of tensionless strings had been linked with the emergence of higher-spin symmetries [4]. The spectrum of tensile string theory includes an infinite tower of massive particles of arbitrary spin. In the tensionless limit, all these particles become massless and the theory is thought to exhibit higher-spin symmetry. In fact, the tensionless limit of string theory has been related to Vasiliev's higher spin theory [5], leading to several proposals of holographic dualities [6][7][8][9][10]. More recently, an exact AdS 3 /CFT 2 correspondence has been proposed [11,12], where the authors gave strong evidences that the tensionless limit of type IIB string theory on AdS 3 ×S 3 ×T 4 is dual to the symmetric orbifold CFT Sym N (T 4 ).
While the study of tensionless string in AdS is very interesting and important, especially for the AdS/CFT correspondence, the tensionless (super)string in flat spacetime, which is often called null strings since the early work of Schild [13], has drawn much attention in the JHEP08(2023)133 past few years. In fact, there have been many early efforts towards the quantization of the tensionless (super)string [14][15][16][17][18][19][20]. These works started from Schild's action of the tensionless string [13]. In contrast, the recent studies on the tensionless string are based on the ILST action [21] (or its supersymmetric generalization) of the null string. The advantage of the ILST action is that two-dimensional(2D) Galilean conformal symmetry or equivalently the Bondi-Metzner-Sachs (BMS) symmetry arises as the residual gauge symmetry on the tensionless worldsheet [22]. Since 2D conformal symmetry offers a guiding principle for the construction of the usual tensile strings in the conformal gauge [23], one expects that BMS symmetry will be important for the tensionless string in the same spirit. Moreover, since the BMS symmetry arise as the asymptotic symmetry group of flat spacetime, the tensionless (super)string could be related to the flat-space holography. Besides, the null string turns out to be closely related to the ambitwistor string [24,25] (see also [26]).
As mentioned above, the tensionless string is closely related to the BMS field theories (BMSFT). The BMS field theories is a type of non-relativistic conformal field theories with the following scaling symmetry and boost symmetry, x → λx, y → λy, x → x, y → y + vx. (1.1) The symmetry algebra can be enhanced to the two dimensional Galilean conformal algebra [27], which is isomorphic to the BMS algebra in three dimensions. The generators of GCA 2 (BMS 3 ) include the superrotations L n and the supertranslations M n , satisfying the following commutation relations (1. 2) The algebra is of infinite dimensions, just like the Virasoro algebra in CFT 2 . The BMS 3 /GCA 2 isomorphism motivates a lot of works establishing the holography in asymptotic flat spacetimes, see e.g. [28][29][30][31][32][33][34][35][36]. On its own right, BMSFT has been widely studied in the past few years. These theories are typically not unitary and exhibit novel features [37][38][39][40], especially the appearance of the boost multiplet [38]. Nevertheless, the BMS (GCA) bootstrap seems still viable [38,39,[41][42][43]. Very recently, a few concrete models of BMSFT had been constructed [44][45][46][47], which are in fact closely related to the tensionless string. 1 In this work, we would like to study the tensionless bosonic string and superstring from the path integral point of view. There have been some studies on the tensionless bosonic and superstring from null string point of view [48][49][50][51][52][53][54] since the discovery of BMS symmetry in the ILST action in [22]. In particular, the quantum theory of the tensionless bosonic string had been studied in [52] and three distinct choices of tensionless vacua had been JHEP08(2023)133 found (see also the early work [55] for a systematic discussion of different choices of vacua). Then the lightcone quantisation had been performed for these theories to find the critical dimensions [53]. In the present work, we start with doing the path integral quantization of the tensionless bosonic strings and reproduce the results of critical dimensions in [53]. 2 When doing the path integral, we introduce the BMS bc-ghost, which turns out to be the inhomogenous ultra-relativistic(UR) limit of the usual bc-ghost. This theory is another BMS free theory apart from the BMS free scalar [44] and fermion [45][46][47]. We also find another BMS bc-ghost by taking the homogenious UR limit of the usual bc-ghost. However, this theory is not related to the path integral of the tensionless bosonic string and only contains boost singlet representations.
Furthermore, we discuss the path-integral quantization of the tensionless superstring. The tensionless superstring had been studied in [49][50][51], where two versions of the BMS fermion action were proposed, leading to homogenous and inhomogenous tensionless superstrings respectively. We give a more complete classification of the quantum tensionless superstrings by including different possible amounts of supersymmetries as well as different kinds of gravitino fields. We introduce BMS βγ ghosts into the study and do canonical quantization on different tensionless superstrings, and calculate their critical dimensions by imposing the conformal anomaly cancellation condition. We find that not all of these theories are consistent because the critical dimensions in some cases are not integers. The non-integer critical dimensions do not appear in the usual tensile superstrings, so these tensionless superstrings with non-integer critical dimensions can not be obtained by taking the tensionless limit of the tensile superstrings. Moreover, the BMS bc and βγ ghost field theories present novel examples of BMS free fields and deserve more careful studies. We show that there exist underlying enhanced symmetries in these theories, which are generated by BMS-Kac-Moody algebras. In particular, they include an anisotropic scaling symmetry. This is similar to the BMS free scalar and fermion [45].
It would be illuminating to compare our results with the ambitwistor (super)strings. Classically, the ambitwistor string is a gauge-fixed version of the null string [24]. At the quantum level, this equivalence holds when quantizing the null string in the flipped vacuum or "normal-ordering prescription" (terminology used in [24]). Relations between tensionless superstrings and ambitwistor strings had also been discussed in [56,57], at the level of chiral superstring integrands. The N = 2 ambitwistor superstring had been discussed by Mason and Skinner [58], 3 and its critical dimension was found to be d = 10. For the ambitwistor strings with different possible amounts of supersymmetries [59], their critical dimensions in fact coincide with our results of homogeneous tensionless superstrings (see table 1). 4 This coincidence reflects the equivalence between the ambitwistor (super)strings and the null (super)strings [24]. Besides, our study includes more types of tensionless superstrings, whose critical dimensions are given in table 2 and table 3. 2 More precisely, only the results of the induced vacuum and the flipped vacuum can be reproduced. The critical dimension of the oscillating vacuum is not accessible by the cancellation of the BMS conformal anomaly. 3 In [58], they also discussed the bosonic and N = 1 cases. 4 According to the result c m = 52 − 11N below eq. (3.5) in [59], it is easy to find exactly the same critical dimensions as in table 1.

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Outline of the paper. The remaining parts of this paper are organized as follows. In the next section, we discuss the path-integral quantization of tensionless bosonic string and introduce BMS bc ghosts. In section 3, we turn to the study of the path-integral of tensionless superstring. We construct the actions of BMS βγ ghosts for various tensionless superstrings, and compute the critical dimensions using the canonical quantization. In section 4, we investigate the underlying symmetry algebra of BMS bc and βγ ghosts. We end with some discussions in section 5. In the appendix, we use the lightcone quantization to compute the critical dimension of the inhomogeneous doublet tensionless superstring, as a consistent check.

Tensionless bosonic string and BMS bc ghosts
In this section, we discuss the path-integral quantization of the tensionless bosonic string. Similar to the usual tensile string, we need to introduce the BMS version of bc-ghost for the Faddeev-Popov determinant. We will show that this intrinsic BMS bc-ghost can be obtained by taking the inhomogeneous ultra-relativistic limit of the usual bc-ghost. By contrary, we can obtain the homogeneous BMS bc-ghosts by taking the homogeneous UR limit as well, but find that they have nothing to do with the path-integral of the tensionless bosonic string.

Path-integral and the BMS bc-ghost
We start with the ILST action [21], Here V is a vector density of weight − 1 2 describing the geometry as The partition function is where ξ represents the BMS transformation keeping V invariant, V ξ ≡ V + δ ξ V = V , and it is given by , In the partition function, Dξ is the integration over different geometries of the worldsheet and DV is the integration over the degrees of freedom in the diffeomorphism. We may transform the integration DV into the integration of an intrinsic diffeomorphism parameter by using the following identity

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where V is the diffeomorphism transformation of V 0 and we have chosen the gauge to set V 0 = (1, 0). Inserting eq. (2.5) into the partition function, we obtain If there is no anomaly for diffeomorphism, i.e. DV DX = DV DX , it turns out that (2.7) In the last step V = V gauge has been imposed. If there is no anomaly for Weyl (BMS) transformation parameterized by ξ, then For simplicity, we focus on the case that the worldsheet is a null-cylinder, and have (2.10)

BMS bc-ghost action and its symmetry
Now we can introduce the BMS bc-ghost system to take into account of the Faddeev-Popov Here we want the action for the bc-ghost to have the symmetry of diffeomorphism, which requires that the whole weight must be −1. We choose to include an additional vector density V in the action, since the operator ∆ β α already has a weight − 1 2 . Then c α is a fermionic vector and b βγ is a fermionic tensor, and they form a multiplet, as we will show in section 4.
Another problem is that we only need two independent components for b-ghost. A simple way to solve the problem is making the tensor b symmetric, i.e. b αβ = b βα . As b 11 component does not appear in the action, there are exactly two indenpendent components JHEP08(2023)133 in b ghost. Note that we cannot make the tensor b traceless because the traceless condition cannot be preserved under the BMS transformations. For convenience, we define With the bc-ghost, the partition function becomes (2.14) It is invariant under the BMS transformation (2.15) or more explicitly, (2.16) The corresponding Noether current and the conserved charge are, respectively, (2.17) From Q one can read the stress-tensor, In addition, one can check that any field Φ fulfills the equation JHEP08(2023)133

BMS algebra
From the action (2.14), we get the equations of motion which allow us to read the mode expansions (2.22) From canonical quantization, we get the (anti-)commutation relations as In addition, we can re-combine the coefficients in the mode expansions of X µ by then we have Substituting the mode-expansion into the expression for T 1 and T 2 , we find

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By using the Fourier transformation, we obtain the generators which generate the BMS algebra (2.28) Here A L and A M are the anomaly terms, which include the contributions from the central charges and the normal-ordering terms.
If we want all the anomalies to vanish, the critical-dimension and the normal-ordering constant must satisfy certain conditions. This is only available for the flipped vacuum and the induced vacuum, because in the oscillating vacuum, most generators acting on the vacuum give infinite number of states and there is no appropriate way to regulate the resulting infinity.
In the induced vacuum, which is annihilated by all non-zero B modes, i.e.
there is no constraint on the critical dimension, which means if only the normal-ordering term a = 0. For the flipped vacuum which is in the highest-weight representation, it is defined by (2.31) Then we have the anomaly terms where D is the dimension of the target-space of the string. Introducing the normal-ordering term a L , we find

BMS bc-ghost from the UR limit
In this subsection, we show how to obtain the tensionless bc ghost by taking inhomogeneous ultra-relativistic(UR) limit on the tensile string. In the usual tensile string theory including the bc ghost, we have the action keeping in mind that b is symmetric and traceless, The bc ghosts in the tensile string theory have the mode expansions as 5 with the anti-commutation relations We may take the following UR limit 38) and find that the action becomes (2.39) Before considering the UR limit of bc ghosts, we should consider the BMS transformations in the limit carefully. The original diffeomorphism takes the form

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where (2.41) Under the UR limit, the transformation conditions become Similarly, considering the BMS algebra as the UR limit of two copies of Virasoro algebra carefully, we find This suggests that we should not neglect O( 2 ) terms in taking the UR limit.

Inhomogenious limit.
There are different options for the bc-ghost fields in the UR limit. Let us first consider the inhomogenious limit, which requires that with other components remaining the same. If we define we reproduce the theory discussed before through intrinsic analysis The transformation rules for the fields can be obtained by taking the UR limit as well.
Comparing the mode expansions before and after taking the UR limit, we find Homogeneous limit. Next, we turn to another option for the bc-ghost fields in the UR limit. In this so-called homogeneous limit, all the fields rescale in the same way 48) and the action for the ghost is

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we obtain a theory with action which is quite different from the theory we studied before. From the equations of motion, we can get the mode expansions of the fields. For the field X µ , it is the same as the inhomogeneous case. But the mode expansion for the ghost fields are different, (2.52) The canonical quantization leads to the anti-commutation relations Comparing the mode expansions before and after the UR limit, we find With the mode expansions, we obtain the BMS generators (2.55) The Virasoro generators take the same forms as the ones in the inhomogeneous case, while the generators M n are slightly different. We find the same critical dimension and the normal-ordering constant.
BRST charge. Here we demonstrate that only the inhomogenous bc ghost is the correct ghost system for the tensionless string. Actually, the intrinsic analysis implies that there must be a derivative with respect to the coordinate σ in the action for the ghosts, but the homogeneous bc-ghost fields do not contain such a term. More explicitly, we can check the BRST charge, and we find that only in the inhomogenous case, the UR limit of the BRST charge of original tensile string theory is meaningful. Let us start from the BRST charge of original tensile string which satisfies the relation

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Then we can consider different limits. It turns out that the Virasoro generators are related to the generators of BMS algebra in the following way, In the homogenous case, the relation (2.58) holds only for the fields X µ . But in the inhomogenous case, the relation (2.58) holds not only for the fields X µ but also for the bc-ghost generators. After calculations, we obtain a finite BRST charge for the inhomogeneous case This BRST charge can also be expressed in terms of BMS fields defined on the null plane as (2.61) Then we can use the operator product expansion(OPE) to check its nilpotency which is obviously correct if the anomaly vanishes since the UR limit works well. In fact, the UR limit of eq. (2.57) gives us (2.62) In the homogenous case, the relation (2.58) does not hold for the bc-ghost generators. In this case, after some calculations, we obtain a divergent BRST charge which suggest that the homogenenous bc-ghost does not make sense.

Tensionless superstrings and BMS βγ ghosts
In the previous section, we have discussed bosonic tensionless string and the corresponding bc ghosts, while in this section we turn to the fermionic tensionless strings and the corresponding βγ ghosts. We discuss in detail in section 3.1 the taking-limit procedure from tensile JHEP08(2023)133 fermionic string, and find that different rescalings of the fermion field components result in homogeneous and inhomogeneous fermionic string actions. For every type of fermionic string, we have two different rescalings of gravitino field components, and we obtain four tensionless superstring actions. Using the Faddeev-Popov trick, we introduce the corresponding βγghost field actions to fix the gauge degrees of freedom of the gravitino field. Further in section 3.2, we use the canonical quantization to read the critical dimensions of the four types of tensionless superstrings. Strangely, the critical dimensions of two types of tensionless superstring are not integers.

Tensionless fermionic strings and βγ-ghost
In the literature, the tensionless fermionic string was constructed by taking the UR limit from relativistic superstring [49][50][51]. As in the bosonic case, we can get the βγ ghost of the tensionless fermionic string by taking the tensionless limit from tensile βγ ghost. However, similar to the bc-ghost case discussed before, there is ambiguity in determining the scaling behavior of the ghost fields. Thus we prefer to construct the βγ ghost intrinsically, where we can uniquely determine the ghost Lagrangian.
In this section, we try to construct βγ-ghost fields intrinsically by cancelling the gravitino degrees of freedom. It has been found that in taking the tensionless limit, two different rescalings of the fermions lead to two kinds of fermions, i.e. homogeneous and inhomogeneous ones, and two rescalings of the gravitinos. Together, there are four kinds of βγ-ghost Lagrangians.
Let us first briefly review the construction of βγ ghost for the tensile fermionic string. The Lagrangian of the superstring in a curved spacetime is where the Greek letters α, β, . . . = 0, 1 are the worldsheet indices. The gravitino field χ α actually carries the spinor index χ α = χ αA with the capital letters A, B, . . . = 0, 1 being the spinor indices, and thus it contains four components, i.e. χ 00 , χ 01 , χ 10 , and χ 11 . There are two fermionic symmetries for this Lagrangian, one being the super symmetry(SUSY) transformation and the other being super-conformal transformation where η is a Majorana fermion. The total number of degrees of freedom in these transformations is four, which just fixes the degrees of freedom of χ. The fixing is done by inserting in the path integral the identity

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where χ is the transformed gravitino under a SUSY transformation generated by and a super-conformal transformation generated by η, and similar for X and ψ in the following equations. The Jacobian δχ δ( ,η) can be rewritten in terms of the path-integral of the ghost field(s) f ghost : Thus the generating function of the superstring is where we drop an infinite overall coefficient. The Faddeev-Popov determinant arose in fixing χ is compensated by introducing the βγ fields into the action. For tensionless fermionic string in different limits, the discussion will be slightly different, as we will show case by case.
In [60], the tensionless limit of string theory was proposed. For the fermion with two components, there is a degree of freedom in taking the tensionless limit, as the relativistic rescalings on two components could be same or different. Different rescalings correspond to homogeneous and inhomogeneous sectors, respectively. Here we apply the ILST trick for the fermionic string. The action of a tensile string is where h αβ is the inverse of the metric on the worldsheet, and h = |det h αβ |. Obviously, taking T → 0 limit directly in (3.7) gives nothing interesting. The ILST trick [60] is to introduce two Lagrange multipliers λ and ρ such that the metric becomes where we denoteT = 2λT for simplicity. Thus the action becomes and we are allowed to take the T → 0 limit. To consider the fermions, we further introduce the inverse of the zweibein e α a , which have one Lorentzian degree of freedom parameterized by |cosh θ| = |a| ≥ 1,

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Note that we only define the inverse of the zweibein, not the zweibein itself, since when we take the tensionless limitT → 0, the inverse of the zweibein is degenerate. The Dirac gamma matrices ρ α in curved space are related to the ones ρ a in flat spacetime by The tensile fermionic Lagrangian is where the fermion is Majorana fermion, i.e. ψ † A = ψ A with capital Latin letters A, B, . . . label the spinor index. In the following, we omit the spacetime index µ and set the Lagrangian multiplier ρ = 0 for simplicity, and thus have In taking the tensionless limitT → 0, we should keep tracking of the rescaling of the components of the fermion, (3.14) It should be noticed that there is also a rescaling factor for the Lorentzian rotation generator in spinor representation: where S ψ is the representation matrix of the rotation S acting on ψ. In order to keep both components of the fermion ψ in the tensionless limit, the scale factors must obey the relation which have two independent solutions, corresponding to the homogeneous limit with |s 0 − s 1 | = 0 and inhomogeneous limit with |s 0 − s 1 | = 1. On the other hand, if we impose |s 0 − s 1 | > 1, one of the fermion component (say ψ 1 ) vanishes in taking theT → 0 limit, and we get a trivial Lagrangian This trivial Lagrangian can also appear in the reduction of homogeneous superstring, and the corresponding βγ ghost will be discussed later.

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With the parameter a = 1, the gamma matrices are (3.19) and the Lagrangian is L As the two components of the fermion are independent with each other, it is possible to turn off either one of them, leading to the trivial Lagrangian (3.17).
The two components of the fermion in the homogeneous limit are in a singlet representation (3.18), thus we should set two components of in the SUSY transformation also in a singlet representation. Plugging in the gamma matrices (3.19), we know that the α = 1 components of χ do not contribute to the Lagrangian and the SUSY transformation. Besides, for the tensionless Lagrangian, the super-conformal transformation is now Thus there are exactly two degrees of freedom in χ in the Lagrangian to cancel, while there are two degrees of freedom of in the SUSY transformation. Using the fact that we immediately read the Lagrangian of βγ ghosts where the second equality is the expression in the flat space gauge. γ A is a bosonic spinor field, and β αA is a bosonic vector-spinor field whose α = 1 components do not contribute.
With the bosonic sector together, we have the Lagrangian of homogeneous tensionless superstring: As mentioned before, we can turn off ψ 1 to get a trivial Lagrangian (3.17). Correspondingly, we can turn off β 01 and γ 0 to get its ghost Lagrangian Thus the full trivial tensionless superstring Lagrangian is where δ 1,A is Kronecker delta function. However, the standard treatment of the inverse of the zweibien leads to a trivial Lagrangian. To get the inhomogeneous Lagrangian, we have to do analytic continuation on the parameter a such that it could take any real value a ∈ R rather than |a| ≥ 1. And finally we take a = 0 to get In fact, extending to a = 0 lead to purely imaginary zweibien and a real gamma matrix ρ 1 .
Notice that the flat gamma matrix ρ 0 is untouched in this extension and remains still pure imaginary. The extra imaginary factor i introduced by this extension should be taken out by hand, such that for Majorana fermion ψ, the Lagrangian is real Similarly, there is ambiguity in the relative rescalings on different components in χ. There are two different rescalings, corresponding to two representations of the rotation S, The corresponding SUSY transformations and super-conformal transformations are different. Let us discuss them case by case.
Vector-doublet gravitino. In this case, χ (d) 11 dose not contribute to the Lagrangian thus the Lagrangian only contains three χ components. The Lagrangian in the flat gauge is and the SUSY transformation is formally the same with (3.2). Furthermore, the superconformal transformation is Thus the three degrees of freedom in these two transformations just fix all components of χ field. Denoting the parameter in the super-conformal transformation as η 0 , we find that the composite transformation is

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Now the Faddeev-Popov determinant (3.6) can be expressed as the ghost field path-integral from which we read the Lagrangian of the βγ-ghost field with β 01 = β 10 . Therefore we obtain the full Lagrangian of the theory, after including the one of bc-ghost field, We will refer to this theory as the inhomogeneous doublet tensionless superstring.
Vector-singlet gravitino. For the gravitino in the vector-singlet representation, only χ (s) 01 contributes to the Lagrangian and the SUSY transformation. In the flat gauge, we have We only need one parameter 1 in the singlet representation under rotation S for SUSY transformation: we get that the ghost Lagrangian is simply The theory with its ghost will be referred to as the singlet theory, and the full Lagrangian is

Critical dimensions for different tensionless superstrings
In the last sub-section we have obtained four kinds of tensionless superstring theories, the homogeneous superstring (3.24) and its trivial reduction (3.26), the inhomogeneous doublet superstring (3.38) and inhomogeneous singlet superstring (3.44). In this section, we use the canonical quantization to calculate the critical dimensions of these string theories. It turns out that the canonical quantization is suitable for the flipped vacuum |vac f , which is BMS invariant: In the following discussions, we will simply focus on the flipped vacuum, and find that the critical dimensions of the homogeneous superstring and the doublet inhomogeneous superstring are the same as the usual superstring, but the ones of the trivial homogeneous superstring and the singlet inhomogeneous superstring are not integers.

Homogeneous and trivial tensionless superstring. The action of the homogeneous case in the flat gauge is
From the equations of motion, we can read the mode expansions of the fields (3.47) The canonical quantization leads to the following nonvanishing commutation relations among the modes (3.48) The stress tensors give the conserved Noether current of the diffeomorphism. After using the equations of motions, they have the following forms Thus the corresponding conserved charges are The SUSY transformation on the fields take the following forms (3.51) The corresponding conserved supercurrents are The conserved charges are defined as which can be expressed in terms of the modes Considering the following two expectation values in the flipped vacuum And from the expectation values  As we showed before, the above action can be further simplified by setting half of the fermionic fields vanishing. The resulting tensionless trivial superstring has the following action in the flat gauge: (3.60) Compared with the action (3.60), the action (3.46) could be viewed as the one for a N = 2 superstring theory, while the trivial one could be taken as an N = 1 theory. One may construct the theories with more supersymmetries as well. In all these cases, the discussion  is similar and straightforward. However, in the case of trivial tensionless superstring, the critical dimension now is not an integer, which implies this superstring theory is not a good one. In fact, the total anomalies of NS-and R-sectors in the theories with N supersymmetries are, respectively, Inhomogeneous doublet tensionless superstring. The action of the inhomogeneous doublet superstring theory in the flat gauge is (3.62) The mode expansions of the fields are (3.63) with the commutation relations among the modes

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(3.64) Similar to the homogeneous case, we can read the conserved charges from the Noether currents, corresponding to the diffeomorphism and supersymmetric transformations. In the end, we find (3.67)

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Similar to the homogeneous case, considering the expectation values And from the expectation values R sector:  Inhomogeneous singlet tensionless superstring. The action of the inhomogeneous singlet superstring in the flat gauge is (3.72) Obviously, only the βγ part is different from the one in the doublet theory, and we only need to study this part. The mode expansions are   Table 3. Critical dimensions of inhomogeneous singlet tensionless superstring theories.
Similar to the discussions above, we can find and

BMS free theories: bc and βγ ghosts
As we shown in previous sections, in the path integral formulation of the tensionless (super)string, we need to introduce the BMS bc ghosts and βγ ghosts to account for the Faddeev-Popov determinants. These two kinds of ghosts can be obtained by taking the inhomogenous UR limits of the usual ghosts as well. In this section, we are going to study the properties of BMS bc and βγ field theories in more details.
In fact, the BMS bc and βγ field theories present new kinds of BMS free field theories. The other BMS free theories, the BMS free scalar and free fermions, had been studied in the literatures [44][45][46][47]. These theories exhibit novel features such as the staggered JHEP08(2023)133 modules and enhanced underlying symmetries, besides the expected appearance of the boost multiplets [38]. Here we aim to investigate if these novel features persist in the BMS bc and βγ field theories as well.
In the BMS free fermions, it has been shown that the underlying symmetry algebra is a BMS-Kac-Moody algebra with a U(1) Kac-Moody subalgebra [45]. As we will show in the following, the underlying symmetry of the BMS bc ghosts is also generated by a BMS-Kac-Moody algebra, but now the Kac-Moody subalgebra is built from a threedimensional non-abelian and non-semi-simple Lie algebra. Besides, We will show that the symmetry algebra of the BMS free scalar also contains a non-abelian and non-semisimple Kac-Moody subalgebra.
Unlike the discussion in previous sections, here we will study the ghost theories defined on the plane, whose coordinates are denoted as x and y. Our discussion will be sightly more general to include BMS ghosts with general scaling dimensions. Because the discussion of the BMS βγ ghosts is quite similar with the BMS bc ghosts, we will mainly focus on the BMS bc ghosts.
We start with the action of the BMS bc ghosts In this theory, the fundamental fields b ≡ (b 1 , b 0 ) T and c ≡ (c 0 , c 1 ) T form two rank-2 boost multiplets. Their conformal dimensions and boost charges are, respectively, This can be easily seen from the infinitesimal transformation law of the field b 1 , b 0 , c 0 , c 1 in (2.16). Recall that a rank-r multiplet has the following behaviour under a finite BMS transformation [38,44]:Õ where O a (x, y), a = 0, 1, . . . , r − 1 is the a-th primary field in the multiplet, and is a finite BMS transformation. From the transformation law (4.3), one can get the infinitesimal form of the transformation. In particular, for the rank-2 case, the transformations turn out to be where ≡ (σ) and ω ≡ ω(σ, τ ) = (σ)τ + υ(σ) are some infinitesimal transformations. Identifying , ω with ξ 1 , ξ 0 in eq. (2.16) and noticing the identity ∂ τ ω = , we find that b, c are both rank-2 multiplets with quantum numbers (4.2).

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As usually do in studying 2D conformal field theory, we would like to do the analysis on the plane. Through the plane-cylinder map x = e iσ , y = iτ e iσ , (4.6) and making use of the transformation law (4.3) for the multiplets b, c (here f (σ) = e iσ , g(σ) = 0), we find the action of bc ghosts on the plane has a similar form with the one on the cylinder: Besides, we want to study more general BMS bc-ghosts, whose conformal dimensions and boost charges are Note that the above quantum numbers still keep the invariance of the action (4.7) under the BMS transformations, and when η = 2 they return to the ones of the BMS bc-ghosts arising from the path-integral of the tensionless string.
One can easily work out the canonical quantization, just as what we have done on the cylinder (and for η = 2). The mode expansion now becomes: Using the anti-commutation relation (4.10), the vacuum condition (4.11) and the hermitian relation, we find the following correlators, M (x 1 , y 1 )M (x 2 , y 2 ) ∼0. (4.16) This is just the form of the stress tensor OPE in a general BMS field theory. From above, we know that the central charges are: .

Enlarged BMS symmetry
Apart from the BMS symmetry, there is a ghost number symmetry G , just as in the usual bc ghosts. It is realized as 18) or in a finite form The corresponding conserved current is Note that the transformation (4.19) is unique in the sense that once the rescaling behaviour of any one of the ghost fields is determined, the ones of the remaining three fields are determined accordingly. However, in the usual bc ghosts there are two independent such kind of symmetry transformations: the ghost and anti-ghost number symmetries. Notice the fact that the BMS bc ghosts are the inhomogenous UR limit of the usual bc ghosts, it is expected that there will emerge another symmetry. This symmetry turns out to be: The corresponding Noether current is: Note that J y G = J x G . The symmetry G λ and G θ can be enhanced to local ones: We would like to comment on the ghost number symmetry here. In the usual string theory, the ghost number symmetry of the bc ghosts could be anomalous when we put them on a general curved background. In JHEP08(2023)133 the case of the tensionless string, similar anomaly may appear as well. Nevertheless, here we focus on the BMS bc ghost filed theory defined in a flat space-time, we have not found any anomaly for the ghost number symmetry, similar to the bc ghost in the usual string theory in flat spacetime.
In fact, the symmetry of the BMS bc-ghost system is even larger. In [45], it was shown that in the BMS free scalar and fermion, there is an anisotropic scaling symmetry. This kind of symmetry also appears in the BMS bc-ghost system. It is easy to see that the action (4.7) is invariant under the following scaling transformations, which keeps x invariant, (4.24) Note that different from the BMS free fermion or scalar, the scaling behaviour of the ghost fields is characterized by a parameter α. As we will see, this is in fact due to the existence of the ghost number symmetry G . One can also find another scaling symmetriesD α , which keeps y invariant, This symmetry is not an independent one becausẽ where D is the isotropic scaling symmetry in the BMS algebra So in the following we need only to study D α . It is easy to derive the corresponding conserved Noether currents of D α : (4. 28) In fact, among all D α s, there is only one independent one, say D α 0 , and other symmetries are constructed as combinations of D α 0 and the ghost number symmetry. To be precise, 6 Furthermore, among all the scaling symmetries, there is actually a natural one with α 0 = −η. It is natural in the sense that only when α 0 = −η, the OPE among the current J x D −η and the stress tensors gives the BMS-Kac-Moody (sub)algebra generated by L n , M n , J 3 n in (4.48), which coincide with the space-time BMS-Kac-Moody algebra (4.34) up to central terms. From (4.24) and (4.27), one find that when α = −η, the dimensions of b 0 (or c 1 ) under D α JHEP08(2023)133

The full symmetry algebra
Now we derive the underlying symmetry algebra. In all, there are five currents encoding the symmetries T (x, y), M (x), J 1 (x, y), J 2 (x), J 3 (x), (4.36) which have the modes expansions (4.37) Note that in the above, the y dependence comes from the equations of motion. As L n = (−n − 2)M n ,J 1 n = (n + 1)J 2 n ,J 3 n = −η(n + 1)J 2 n , there are in total five kinds of generators in the algebra: L n , M n , J 1 n , J 2 n , J 3 n . where the structure constant f ab c has only non-vanishing components 40) and the level matrix k ab is of the form (4.41) Note that the matrix of the central terms is degenerate. This is in fact due to the degeneration of the Killing form of the underlying Lie algebra. In other words, this algebra is a non-semisimple Kac-Moody algebra. The underlying Lie algebra is a three-dimensional non-semisimple (non-abelian) Lie algebra generated by e 1 , e 2 , e 3 with commutation relations  It has a maximal abelian ideal generated by {e 1 , e 2 }. This algebra is actually a direct sum of R (generated by e 1 ) and a 2-dimensional non-abelian Lie algebra 8 (generated by {e 2 , e 3 }).

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Remarkably, these two subalgebras no longer decouple when being lifted to the Kac-Moody algebras: k 13 does not vanish in (4.41). Notice that the (global) BMS algebra itself is a non-semisimple Lie algebra, thus it is not strange that non-semisimple Kac-Moody algebra appear in BMS field theories. It is worth pointing out that a contraction of two Kac-Moody algebras always leads to non-semisimple Kac-Moody algebras. 9 This is similar with the BMS algebra, which is the contraction of two Viraroso algebras and is non-semisimple. Interestingly, the Kac-Moody algebra appear here can not be any contraction of two other Kac-Moody algebras.
In fact, non-semisimple Kac-Moody algebra appears in the BMS free scalar theory as well. In [45], it was shown that there are two currents with ∆ = 1: the one associated with the translational symmetry: φ → φ + Λ(x), and the other one associated with the anisotropic scaling symmetry, The corresponding two currents have the following forms: From the OPE of these currents, it is easy to find the underlying Kac-Moody algebra. It turns out to be: [j a n , j b m ] = g ab c j c n + nκ ab δ n+m,0 , (4.45) where the structure constant g ab c has only non-vanishing components 46) and the level matrix κ ab is of the form This Kac-Moody algebra is not semisimple and can not be a contraction of any two other Kac-Moody algebras as well. Its underlying Lie algebra is a two-dimensional subalgebra of (4.42) generated by (e 2 , e 3 ). Now we turn back to the symmetry algebra of the BMS bc ghosts. We have five sets of generators in (4.38): three of them (J 1 n , J 2 n , J 3 n ) form a Kac-Moody subalgebra, two of them (L n , M n ) form the BMS algebra. Making use of the OPE of the five currents, we can read the full symmetry algebra: 9 The contraction of two u(1) Kac-Moody algebras had been studied in [61]. Non-abelian contractions had been studied recently in [62].

Inhomogenous BMS βγ ghost
Another ghost field theory we obtained from the tensionless BMS fermionic string is the BMS βγ system. The action (on the plane) is: Since the structure of this theory is very similar with the BMS bc ghost system, we will not give a detailed analysis but just give some brief comments on the BMS βγ ghosts. It is easy to see that this theory contains two rank-2 boost multiplets: γ = (γ 0 , γ 1 ) T , β = (β 0 , β 1 ) T . They have the following dimensions and boost charges: One can also study more general BMS βγ ghosts with dimensions: Then the stress tensors of the generalized theory are This theory has the following central charges: Similar to the BMS bc ghosts, the BMS βγ ghosts also has the anisotropic scaling symmetry [45], thus all BMS free theories have this symmetry. It turns out that the full symmetry algebra of the BMS βγ ghosts is the same as the one of the BMS bc ghosts (up to central charges).

Conclusion and discussion
In this work, we studied the path-integral quantization of tensionless superstring. We showed how to introduce the BMS bc and βγ ghosts intrinsically to account for the Faddeev-Popov determinants appeared in the path integrals. For the bosonic string, the BMS bc-ghosts can be obtained by taking the inhomogeneous ultra-relativistic limit of the usual bc-ghosts.
For the tensionless superstring, we found four kinds of βγ-ghosts via intrinsic analysis. We did canonical quantization of the ghost systems and found the critical dimensions of different tensionless superstrings. It turned out that the critical dimensions of the homogeneous superstring and the doublet inhomogeneous superstring are the same as the usual superstring, but the ones of the trivial homogeneous superstring and the singlet inhomogeneous superstring are not integers. We furthermore investigated the BMS ghost systems by taking them as novel kinds of BMS free theories. We showed that the BMS ghosts have enhanced underlying symmetry, just like the BMS free boson and fermion. The underlying symmetry is generated by a BMS-Kac-Moody algebra, with the Kac-Moody subalgebra being built from a three-dimensional non-abelian and non-semi-simple Lie algebra.

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The study in this work, together with the ones on the BMS free scalar and fermion [44,45], suggests that all BMS free field theories have an enhanced space-time BMS-Kac-Moody symmetry. It would be interesting to see whether this enhancement persists in interacting BMS field theories. Besides, with the BMS bc and βγ ghosts in hand, it is possible that one can find the bosonization of them.
Before we conclude our work, we would like to discuss the spectrum of tensionless superstring. As discussed in detail in [52], tensionless bosonic strings have three vacua, which are induced, oscillator and flipped vacuum. The spectrum in the induced vacuum is just the same as the tensile string theory with the tension being taken to zero. In the oscillator vacuum, it is hard to impose the physical conditions. The flipped vacuum has well-defined physical conditions but with the spectrum being truncated. The same things happen in the tensionless superstrings. And the spectrum of homogenous tensionless superstrings has similar structures of tensionless bosonic strings, as shown in [18,49]. Thus we here focus on the spectrum of inhomogenous tensionless superstring in the flipped vacuum.
Due to the non-trivial contraction of super-Virasoro algebra in the inhomogenous case, the left-mover and right-mover are coupled with each other, and there does not exist the NS-R or R-NS sector. This feature of inhomogenous theory implies the absence of spacetime supersymmetry.
In the critical dimension D = 10, the spectrum of the excitations in the flipped vacuum is truncated at N b +Ñ b + N f +Ñ f = a L where a L = 1 for the NS-NS sector and a L = 0 for the R-R sector. Here we have defined that χ r ≡ 1 2 (2ψ r +ψ r ) ,χ r ≡ 1 2 (2ψ r −ψ r ) .

(5.2)
There are two kinds of physical states in the NS-NS sector, with p 2 = 0, a · p = 0, However, all these states are null, i.e.
phy 1| phy 1 = phy 2| phy 2 = 0. Finally, we need to use the super-BMS constraints to determine the rest fields. 10 The constraints are 0 = L n = 1 2 m A −m · B m+n + 1 4 r (2r + n)(β −r · γ r+n + γ −r · β r+n ), leading to JHEP08(2023)133 In the flipped vacuum, we have And the mass is determined as where the index "r" take value in half-integers in the NS-NS sector and integers in the R-R sector. And here we want to emphasize that there does not exist NS-R sector or R-NS sector due to the non-trivial contraction of the super-Virasoro algebra. Furthermore, this implies that the inhomogeneous superstring does not admit the spacetime supersymmetry anymore and its Green Schwarz formalism seems impossible to construct. At last, we discuss on the Lorentzian algebra in the inhomogeneous superstring. Since (A.14) Comparing with usual tensile superstring theory, we find that this Lorentzian algebra exactly matches the well-defined UR limit results. This fact implies that the critical dimension and the normal-ordering constant remain the same as usual tensile superstring.
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