Torsional Newton Cartan gravity from non-relativistic strings

We study propagation of closed bosonic strings in torsional Newton-Cartan geometry based on a recently proposed Polyakov type action. We generalize the Polyakov action proposal to include matter, i.e. Kalb-Ramond field and dilaton and determine the conditions for Weyl invariance which we express as the beta-function equations on the worldsheet, in analogy with the usual case of strings propagating on a pseudo-Riemannian manifold. The critical dimension of the TNC space-time turns out to be 25. We find that Newton's law of gravitation follows from the requirement of quantum Weyl invariance in the absence of torsion. We also find that Weyl invariance in the absence of matter requires vanishing torsion. Torsion can be generated in the presence of dilaton, or for certain non-generic choices of the Kalb-Ramond field. Presence of torsion has interesting consequences in the weak gravity limit, in particular it yields a mass term in the Poisson equation for Newton's potential. We also find that the U(1) shifts of the central charge which is a symmetry of the Newton-Cartan geometry at the classical level becomes anomalous at one-loop.

Einstein's realization that gravity stems from geometrization of the Lorentz symmetry is among the greatest achievements in the history of physics. In general relativity, the equivalence principle is guaranteed by endowing spacetime with a (pseudo-)Riemannian structure that ensures the local Lorentz invariance. This profound connection between geometry and gravity is not unique to the laws of special relativity however, as an analogous connection exists also for the Galilean invariance. A covariant treatment of Galilean symmetry was first presented by Cartan [1][2][3] leading to the discovery of the Newton-Cartan (NC) geometry as the underlying structure of classical Newtonian gravity. Subsequent work [4][5][6][7] clarified the algebra of spacetime transformations and its representation theory that underlies the NC geometry. In particular it was shown in [8] that the NC geometry follows from gauging the Bargmann algebra, the U(1) central extension of the algebra of Galilean boosts, translations and rotations. Finally, the structure of the Newton-Carton geometry has been extended to include torsion [9,10], and referred to as the "torsional Newton-Cartan" (TNC) geometry 1 .
A crucial element in this geometry is the presence of the U(1) gauge symmetry that corresponds to the aforementioned central charge and physically related to the conservation of mass. Non-relativistic gravity has recently been studied in the context of non-relativistic effective actions [11], non-relativistic holography [12], post-Newtonian expansions of general relativity [13], and more recently in the context of string theory [14].
In this paper we ask the question whether the TNC geometry can be UV completed in a consistent theory of quantum gravity and take a few first steps in answering this question in the context of bosonic string theory 2 . One of the triumphs of the ordinary (relativistic) string theory has been the derivation of Einstein's equations in the weak gravity limit by demanding Weyl invariance of the world-sheet sigma model [15]. In our case of string propagating on a manifold with local Galilean invariance, we similarly expect that the demand of quantum Weyl invariance on the world-sheet yields Newton's law in the weak gravity limit. This is what we mean precisely by the consistency of the TNC geometry with quantum gravity.
1 See [10] for a discussion on necessity of including torsion in this theory. 2 Eventually one may need superstrings to tame tachyonic instabilities but we expect this be a natural extension of the calculations we present here.
Various proposals to realize the Galilean symmetries in string theory exist in the literature. The Newton-Cartan geometry has, only recently, been embedded in string theory at the classical level, that is at the tree level of the world-sheet non-linear sigma model [14,16,17]. A parallel and separate line of work [18][19][20][21] started by the original paper of Gomis and Ooguri [22] that realized Galilean symmetry in the context of closed string theory in a particular contraction limit, and, continued by the very recent paper [23] that asks the same question we ask here in the context of the Gomis-Ooguri theory 3 .
We will follow the route taken by the papers [14,17] where a Polyakov type action for string propagating in the TNC geometry was constructed. Taking this Polyakov action as our starting point, we extend it to include bosonic target space matter, i.e. the Kalb-Ramond field B µν and dilaton φ, and we determine both the target space and worldsheet symmetries of this action at the classical level. We then go beyond the tree level and construct the worldsheet perturbation theory in string length l s = √ α , assuming that the target TNC space is weakly curved. We then obtain the target space equations of motion from quantum Weyl invariance of the non-linear sigma model proposed in [14] and its generalizations including the Kalb-Ramond and the dilaton fields. We pay special attention to the target space symmetries, and ask which subset of the symmetries of the classical action is preserved by quantum corrections. It turns out it is very hard to construct a path integral measure that preserves the U(1) central charge symmetry of the classical theory and, as a result, we find that the original U(1) symmetry becomes anomalous at the quantum level.
Our paper is organized as follows. We begin, in section II, by reviewing the Polyakov-type action we use for the closed bosonic string moving in a TNC background and then generalize it to include Neveu-Schwarz background matter, i.e. the dilaton and the Kalb-Ramond field. We then discuss how the target space and worldsheet symmetries are realized at the classical level. Section III constitutes the core of our paper. We use a covariant expansion of the TNC background fields to rewrite the action in the form of a perturbative series in quantum fluctuations parametrized by the string length l s . This expansion coincides with the derivative expansion in the target space and we truncate the series at the second order both in derivatives and in quantum fluctuations. Using this quantum effective action at the quadratic level, we then compute the one loop contribution to the Weyl anomaly in the absence of matter. We then show that the equations of motion arising from the vanishing of the beta functions imply a TNC geometry with zero torsion as well as the equations of motion of Newtonian gravity for the gravitational background. All these calculations are first presented in the absence of background matter in section III D. In section II B we introduce matter in the sigma model through a Kalb-Ramond B-field and a dilaton and repeat the background expansion for this matter extended version of the sigma model. We use a particular choice of the B-field background to compute the one-loop Weyl anomaly at the end of this section. Finally in section IV we present a discussion of the results and provide an outlook. Several appendices where we give details of our (quite lengthy) calculations form a substantial part of this paper.
Note addded: We became aware of a paper of Gomis, Oh and Yan [23] on the quantum Weyl symmetry of the non-linear sigma model for the non-critical string theory in the final stage of our work.

II. THE STRING ACTION AND ITS SYMMETRIES
A. The Polyakov action without matter The geometric data of the TNC geometry in the absence of matter fields is encoded in a pair of vielbeins 4 (τ s , e i s ) and a U(1) connection m s collectively referred as the TNC metric complex. The vielbeins e i s define a degenerate spatial metric through h mn = e i m e j n δ ij and it is possible to use the inverse of the square matrix (τ m , e i n ), denoted as (−υ m , e n i ) with υ m τ m = −1 and τ m e m i = 0, to define an independent spatial inverse metric h mn = e m i e n j δ ij . These spatial metrics together with the temporal coframes, τ m and υ m , are subject to a completeness relation δ m n = −υ m τ n + h mr h rn .
Quite conveniently, the TNC geometry with this geometric data can be derived from a higher dimensional relativistic spacetime with an isometry in the extra null direction-which we will denote as the u-direction-via the procedure of null reduction [24]. In particular we consider the TNC manifold to be d+1-dimensional and the relativistic one will be d+2- 4 We will use letters {m, n, ...} to denote curved TNC indices and {i, j, ...} to denote flat TNC indices.
dimensional. The metric of such relativistic spacetimes can always be written as with ∂ u the corresponding null Killing vector. We label indices of the d+2 dimensional space as M = {u, m}. We also define τ = τ m dx m , m = m s dx s with x m the coordinates of the (d+1)-TNC manifold. It is now possible to derive the world-sheet action for a string moving in the TNC geometry [14,17] starting from the ordinary Polyakov action in the relativistic target space (1): where γ is the determinant of the worldsheet metric γ αβ , and where h αβ =h rs ∂ α X r ∂ β X s and τ α = τ m ∂ α X m are the pullbacks of h rs and τ r respectively 5 .
We consider a closed string without winding, i.e. X m (σ 0 , σ 1 + 2π) = X m (σ 0 , σ 1 ) , and with non zero momentum P along X u with the momentum current Following [14] it is possible to rewrite (2) in a dual formulation where the conservation of the momentum current (4) is implemented off-shell through the classically equivalent Lagrangian where A α is a Lagrange multiplier that enforces conservation of P α u = αβ ∂ β η 2πα off-shell and we defined the combinationh The significance of this combination will become clear when we discuss the symmetries of the theory below.
This procedure introduces a novel degree of freedom, a scalar field η on the world sheet.
To see that (5) and (2) are equivalent one uses the equation of motion for η which gives A α = ∂ α χ for some world sheet scalar χ and identifies the latter with the u-direction χ = X u recovering the original Lagrangian (2). Following [14] we introduce the worldsheet zweibein e a α and its inverse e α a = αβ e b β ba , satisfying e a α e b β η ab = γ αβ and e α a e β b η ab = γ αβ , to rewrite the constraints as A final field redefinition yields the Lagrangian where e α ± = e α 0 ± e α 1 . This is the Polyakov-type Lagrangian for a string moving in a TNC geometry proposed in [14]. We further use the constraints to rewrite (9) in a way more convenient for quantization 6 We will examine the quantum path integral defined by this Lagrangian in the rest of the paper, but we will first extend it to include Neveu-Schwarz matter, i.e. the Kalb-Ramond field and dilaton and then discuss the symmetries of this generalized action both on the worldsheet and in the target space.

B. The Polyakov action with matter
It is straightforward to generalize the action (10) to include standard Neveu-Schwarz matter, i.e. a Kalb-Ramond field B mn and a dilaton φ. Let us first consider the B-field. 6 One should think of implementing these constraints inside the Polyakov path integral to ensure equivalence of the quantum path integrals based on the lagrangians (9) and (10).
Once again, to derive the corresponding Lagrangian we can start from its null lifted version. We then obtain the following action by rearranging the terms that follow from the null reduction of the relativistic d+2 dimensional bosonic Polyakov action with the B-field: where we defined Following the same procedure as in [14] described in section II we compute the momentum and implement its conservation off-shell via Making, once again, the field redefinition integration over the worldsheet fields λ ± now impose the constraints we can cast (15) in the Polyakov form where just as in (10)  When the world-sheet is non-flat, in addition to the B-field, it is also possible to include a dilaton contribution of the form where R is the worldsheet Ricci scalar. The Polyakov path integral then involves a sum over world-sheet topologies that is organized in powers of exp(φ) as usual.

C. Symmetries of the Polyakov action
We will now discuss both the global (target space) and the local (worldsheet) symmetries of the world-sheet action (18) and (19).

Space-time symmetries
The fields in the TNC metric complex, without matter, transform under diffeomorphisms ξ, local Galilean boosts λ i , local rotations λ ij and local U(1) gauge transformation σ and the Lagrangian (5) is invariant under these transformations [14]. These transformations are easily generalized in the presence of matter. All in all, the transformations of all the objects that enter the calculations read In particular, the combinationsh mn andB mn defined in (6) and (13) are invariant under local Galilean boosts and local U(1) transformations respectively (in addition to invariance under local rotations of both). Now, it is straightforward to check that the actions based on (18) and (19) are invariant under the diffeomorphisms, local Gallilean boosts, local rotations and local U(1) transformations. For this to happen, first, it is crucial that h αβ and B αβ appear in the combinationsh αβ andB αβ in (18). Second, one should employ the constraints (17) to show invariance under local U(1) and Gallilean boosts.
In what follows, in addition toh mn andB mn defined in (6) and (13), it will prove useful to introduce the following combinationŝ Φ ≡ −υ s m s + 1 2 h rs m r m s (22) that are invariant under local Galilean boost and rotations as one can easily check using (20). They do transform under local U(1) though: Even though they do not appear in the action at the classical level, we have introducedυ m as the local Galilean boost and rotations invariant version of υ m the inverse of τ m , and the target space scalar Φ which will play the role of the Newton's gravitational potential below.
They will become important when we discuss quantum corrections in the theory. We note thatυ m , τ m ,h mn and h mn are subject to the completeness relation δ r s = −υ r τ s + h rmh ms . Finally, we note that because of the non-trivial U(1) gauge transformation of B mn in (20), i.e. δ σ B = ℵ ∧ dσ, the usual definition of the field strength, H = dB will not be U(1) invariant. However, the combination H − m ∧ dℵ is. Thus, if the U(1) invariance is unbroken at the quantum level, then we expect the field strength H always appears in this combination in the beta-function equations.

U (1) B one-form symmetry
In the presence of the Kalb-Ramond field there is also a U(1) one-form symmetry. It is well-known that the transformation where ∂ M is the partial derivative in the target space, is a symmetry of the d+2 dimensional world-sheet action with the relativistic target space.
However, the TNC geometry that is obtained by the null-reduction of the relativistic target space, does in general have torsion. Indeed, for a generic TNC geometry there is a natural choice of connection Γ m rs defined as [10,25] Γ m rs ≡ −υ m ∂ r τ s + with the property that it is compatible with the metric complex τ m and h mn , namely This connection is not symmetric and possesses a torsion component This means that, after null reduction, the resulting TNC geometry with Kalb-Ramond matter has a modified U(1) one-form symmetry of the form: We see that in the TNC geometry ℵ acquires a new local U(1) symmetry, whereas B transforms under a local one-form symmetry. It is now straightforward to check that the action (18) is invariant under (29) upon use of the constraint equations (17). Invariance of (18) under (30) however requires a non-trivial transformation of the worldsheet field η: which is a trivial shift in the quantum path integral where η is path integrated. Therefore, we conclude that the action, at least at the tree-level, is invariant under both the local one-form symmetry Λ m and the new local U(1) symmetry Λ u . The fact that η is charged under the U(1) that comes from the B-field, i.e. eq. (31), is expected as one can think of η as the direction dual to u, [14]. In this sense the gauge fields m and ℵ can be considered as dual to each other.
In passing, we note that the action (18) enjoys an additional symmetry forB mn given by with Ω an arbitrary spacetime function. To show that (32) is a symmetry it is necessary to use the constraint equations (17).

Local worldsheet symmetries
The actions (10) and (18) are clearly invariant under the worldsheet diffeomorphisms.
These symmetries allow us to cast the worldsheet metric in a diagonal form γ ab = e −2ρ η ab where the conformal factor ρ determines the Ricci curvature of the worldsheet R (locally) We will refer to this choice of gauge as the conformal gauge. The reparametrization gaugefixed Polyakov Lagrangians (10) and (18) further exhibit a residual Lorentz/Weyl gauge invariance of the form (as can be checked straightforwardly) for any worldsheet function f ± . For f + = f − the transformation is a local Weyl transformation and for f + = −f − it constitutes a local Lorentz transformation. Once we have used diffeomorphism invariance to go to conformal gauge it is possible to use local Weyl invariance to fix the mode ρ and completely fix the worldsheet metric γ αβ .
The main purpose of our paper is to discuss the fate of these residual gauge invariances at the quantum level, and we will discuss the cases without and with the Neveu-Schwarz matter separately below. Here it suffices to note that, in the case without matter, the condition for invariance of the Polyakov action S(e, λ, X) under the gauge transformations (34) at the classical level takes the form where the energy momentum one form 7 τ c γ and constraint functions C ± are defined as The condition (35) is nothing but a constrained traceless condition for the energy momentum tensor, and from (36) and (37) it is clear that this conditions holds for the Polyakov action (10). The rest of our work will concern the computation of (35) at the quantum level, in particular, at the one-loop level in the perturbative expansion in α . 7 Even though it is possible to define an energy momentum tensor from τ c γ via T αβ = η cd e d α τ c β it is more natural to define the traceless condition in terms of the energy momentum one form.

III. QUANTUM WEYL INVARIANCE OF STRING IN THE TNC GEOMETRY
A. Background field quantization The quantum partition function that follows from the action (10) is defined by the Polyakov path integral 8 . As for the bosonic strings [15], it will be very helpful to introduce the background field formalism to organize the perturbative α expansion to study the quantum properties of the worldsheet sigma model. To this end, we expand the worldsheet where Ψ ≡ {Ȳ m ,Λ ± ,H} below will collectively denote the quantum fields. Using this expansion, the one loop effective effective action Γ[Ψ 0 ] for the background fields can be expressed [26] as a path integral over the quantum fields as whereS[Ψ 0 , Ψ](0) is the O (l 0 s ) term that arises from substituting (38) in (10). In (39) the zweibeins are completely fixed by the Faddeev-Popov procedure, see Appendix B, using the reparametrization invariance and Weyl symmetry. This, in particular, fixes the function ρ.
If the symmetry (34) is to be consistent at the one loop level then any change of ρ should leave the effective action invariant, this means that the Weyl invariance (35) at the one loop level becomes 9 Of course this requirement only makes sense if we specify the measure in the path integral DΨ which we discuss next. 8 It is crucial to include the contribution from the Faddeev-Popov ghosts that come from the gauge fixing but we will not explicitly show them here. The gauge fixing procedure is discussed in detail in Appendix B. 9 We are assuming that the measure of the path integral can be written in a diffeomorphism invariant way.

B. Path integral measure
Depending on the choice of measure we expect certain symmetries discussed in section II C 1 to be broken. The tricky part in the definition of the measure is over the fluctuations δX m : where G mn is some metric that we now specify. Since the local Galilean and rotation invariances are fundamental and we do not want the measure to explicitly break these symmetries we are led to the following general choice for G mn , that is comprised of the invariantsh mn , τ and Φ: Now, one should ask whether the U(1) central charge symmetry is broken by this choice or not. To satisfy U(1) invariance of the measure in general, we only need the variation of G mn to be expressed as a Lie derivative of the metric G mn 10 : Here, we will demand a more strict definition and ask whether δ σ G mn = 0 or not. In appendix G we showed that this requirement is only satisfied by a specific type of U (1) transformations of the form for some arbitrary function F of the space-time coordinates. This means that the beta function equations we obtain in the end are not expected to be invariant under a generic U(1) symmetry but only under this special type. We show that this will indeed be the case in Appendix G . In the following section we show how (40) can consistently be computed using a covariant target spacetime derivative expansion.

C. Covariant background expansion
The goal of this section is to expressS[Ψ 0 , Ψ](0) in a TNC covariant way using an analog of the Riemann normal coordinates. We first note that, sinceȲ m is defined as a difference of spacetime coordinates, it would not transform as a vector under general coordinate transformations, consequently we need to rewrite it in terms of a spacetime vector. This is achieved [26] by considering a geodesic connecting X m 0 and X m 0 +Ȳ m to rewriteȲ m asȲ where Y m is the tangent vector along the geodesic andΓ m rs is a symmetric connection (e.g. symmetric part of the TNC connection) in the background geometry used to construct the geodesic. Equation (45) defines a coordinate transformation in the neighborhood of X m 0 to the set of coordinates Y m , known as Riemann normal coordinates, that satisfy the following whereR m rst is the Riemann tensor constructed from the symmetric connection. This means that the quantum expansion (38) for X m can be written in terms of the vectors Y m : To computeS[Ψ 0 , Ψ](0) we also need the quantum expansion of the non-linear couplings h mn (X) and τ m (X). This can be achieved by first doing a Taylor expansion inȲ m around X 0 and making use of (45) to rewrite such expansion in the following covariant waȳ withD m denoting the covariant derivative in TNC spacetime defined through the symmetric and where () 0 indicates the expression is evaluated at X 0 . We reproduce below the connection for a generic TNC geometry [10,25], explained in II C 2: with Φ defined in (22) and the normalizations are judiciously chosen such that the normalization of the first term in (50) becomes canonical, i.e. it yields the first two terms in the zeroth order action below. To see this one needs to use the identityh mnυ n = 2Φτ m .
The zeroth order action S 0 is now expressed in terms of only flat indices, and we expand it as with S [a] 0 denoting the at O (D a ) in target spacetime derivatives. In detail we find, where we have explicitly broken the covariance by using The variation of the effective action (40) can now be computed perturbatively as where Z F P is the partition function for the Fadeev-Popov ghosts arising from the gauge fixing procedure, see appendix B. Z 0 and 0 denote the partition function and correlation function computed with respect to the action S By dimensional considerations we expect δ ψ log(Z 0 Z F P ) = c T R with c T a proportionality constant 13 . The coefficient c T is independent of the background fields and depends only on the dimensionality of the TNC spacetime. Therefore, as in the case of the ordinary string, 11 The spin connection is not gauge invariant meaning that it should not contribute to the beta functions. 12 This is true as long we work on globally flat worldsheets. 13 At one loop level this is the only contribution to the anomaly proportional to R.
the requirement c T = 0 fixes the dimensionality of the background geometry. This is the requirement of invariance under conformal reparametrizations, hence the quantum consistency of the theory in the absence of extra dynamical fields. We find that the requirement c T = 0 critical dimension of the d + 1 dimensional TNC geometry is found to be The details of this calculation are presented in Appendix D. This result is somewhat expected, as quantum consistency of the ordinary bosonic string sets d + 2 = 26 and we obtain the TNC geometry by reduction of this 26 dimensional background on a null direction.
Nevertheless, it is still a non-trivial result, as we cannot find a simple argument as to why quantization and null reduction should commute.
Assuming critical dimension, the right hand side of (54) can be written as with the corresponding beta functions {β rs , β m , β} given by Here we defined the "acceleration", the Ricci tensor R mn = R t mtn , a 2 = a m a n h mn , and the spatial Laplacian D 2 = h rs D r D s . The details of the computation of (57), (58), and (59) are relegated to appendix F.
with ∆ r m = h rth rm the projector operator along the spatial directions and Φ the Newton's gravitational potential. Equation (61) is precisely the twistlessness condition on torsion, namely τ ∧ dτ = 0, and it is solved by [13] F mn = a [m τ n] .
In particular we find that the torsion would give rise to a mass term in the Newton's law, however this mass term, as well as all components of the torsion F mn are required to vanish by two of the beta-function equations, i.e. β m = β = 0, which yields a 2 = 0. By using positive definiteness of the spatial components in this equation one obtains a i = 0 and by using the orthogonalityῡ m a m = 0 one finds a 0 = 0, this means that the entire acceleration vector hence the torsion vanishes in this case, a m = 0 and F mn = 0. Hence the beta function equations in the absence of matter simplify as The first one is Newton's law and the last two are the other Newton-Cartan equations obtained from the two linearly independent projections of Einstein's equations. In the following section we see how the introduction of matter-described in section II B-in the non-linear sigma model modifies these equations.

E. Weyl invariance at one loop with matter
The beta functions are modified in the presence of a Kalb-Ramond B mn and dilaton φ fields. The classical worldsheet action in this case is obtained in section II B and given by equation (18). With our current approach it is not possible to compute the beta functions in the presence of an arbitrary ℵ field with the exception of the special case ℵ m = Υτ m where Υ is a constant. To facilitate the computation, we first consider the case Υ = 0, that is ℵ = 0. We will also include the dilaton contribution coming from the classical trace of its energy-momentum tensor, although we will not include its contribution coming from higher loop computations-see the Discussion and Appendix E-. To obtain the contribution of B mn (X) to the Weyl anomaly we first need to write its contribution toS 0 in a covariant way.
Repeating the Riemann normal coordinate expansion procedure above, the contribution of the B-field toS 0 is obtained as where the tensor coefficients {Ā,C} can be found in appendix A and are written in terms of the U(1) invariant field strength tensor 14 Applying the flat index decomposition, (51), the B-field contributes at the first and the second order in the derivative expansion ofS 0 . These contributions read where the relevant tensor coefficients {H, Z} are presented in appendix A. When both (70), (53) and the tree level contribution of the dilaton φ, as computed in appendix E, are taken into account the Weyl variation of the effective action, (54), becomes (71) 14 We define the field strength using the symmetric connection instead of the full one making it invariant under the U (1) B one form symmetry. See discussion in section II C 2.
Here we defined Σλ β ≡ λ − e β + + λ + e β − , and β is given as in (59). The new beta function coefficients {β m ,β rs } are given bȳ Notice thatβ m = 0 when we have twistless torsion. The beta function β rs is modified as More details of the computation of (72), (73) and (74) are relegated to appendix F and appendix E. The presence of the dilaton also modifies the beta function β m , which now becomes The vanishing of the beta functions in the presence of the B-field and the dilaton gives rise to the following non-relativistic equations of motion in the presence of matter F mn F rs h mr h ns = 0 (76) We remind the reader that in these equations Φ denotes Newton's gravitational potential and φ denotes the dilaton. In particular we observe that the Kalb-Ramond field and the dilaton appear as source terms in Newton's law. Furthermore, the conclusion of the previous section, that is, the torsion vanishes in the absence of matter fields can now be avoided by turning on the dilaton. See the next section for a discussion on this issue. Finally, we note that these equations of motion are derived for a specific choice of the Kalb-Ramond field, with the null-space component of the B-field (before the null reduction) B um = ℵ m = 0.
This choice was made for simplicity. The only other special case that we can treat with our methods is when ℵ m = Υτ m with constant Υ. This case is worked out in Appendix F and discussed further in the next section. The calculations of the beta function equations are also quite technically involved with many possibilities of error. For example, we needed to adapt the trick of using Riemann normal coordinates in the background field expansion to the TNC geometry, which, a priori contains torsion. Strictly speaking, the Riemann normal coordinates do not exist for a geometry with torsionful connection [27]. We sidestepped this complication by defining the "semi-Riemann normal coordinates" which satisfy the salient features of the ordinary Riemann normal coordinates, e.g. eqs. (46), when only the symmetric part of the connection is used in the definition. Another subtlety arises in switching from the coordinatesȲ m to Y m in equation (45). This change was necessary to make sure that the difference between the full coordinates X m and the background ones X m 0 transform properly as a vector. We defined this vector as the derivative tangent to a geodesic that connects the two. However, there exists a subtlety in the definition of a geodesic in the presence of torsion. In particular the notion of minimizing an invariant length and the notion of parallel transport by Lie dragging do not yield the same geodesic equation in the presence of torsion. In our calculations we adopt the latter definition, in term of the Lie derivatives, as they are insensitive to torsion. A few words on torsion in the TNC background. Torsion is expected to be absent when there are no matter fields [28], hence our result in eq. (63) is consistent. As mentioned above, one of the beta-function equations both in the absence and presence of matter turns out to be the twistless torsion condition, equation (61) with solution F = a ∧ τ where a = 2υ r F rm is the acceleration [13]. Inserting this in the β m = 0 equation then requires vanishing of torsion. On the other hand, one generically expects to generate it by the Kalb-Ramond field. This is indeed what we find in equations (76-81) but with an interesting addition: it seems that turning on the dilaton is necessary for a non-vanishing torsion. This is easy to see from eq. (78): setting φ = 0, the steps mentioned above in the absence of matter would again yield a m = 0 even when the Kalb-Ramond field is nontrivial. In passing we note that, the presence of φ is necessary but not sufficient for non-vanishing torsion, as one can still solve (78) by setting a m = 0. This finding, that torsion necessitates dilaton, is somewhat surprising as we would expect that the B-field, not the dilaton, generated torsion. This could perhaps be explained if the consistency of the beta-function equations, including the O(l 2 s ) contributions to the dilaton beta function -computation of which requires two loop diagrams that is beyond the scope of this paper -altogether, yield an equation schematically of the form H 2 ∝ (Dφ) 2 + D 2 φ. This is plausible but we cannot directly check this.
One possible way to relax this need of dilaton to have torsion is to allow for more general matter than we did in section III E. Here, we have chosen the null-space component of the B-field before reduction, i.e. ℵ m = B um to vanish for simplicity. The only other case we were able to carry out the calculation of the beta functions was the specific choice ℵ m = Υτ m with constant Υ. The calculation in this case is detailed in Appendix F. As one can see from (F4) in the special case of Υ = ±1 twistlessness condition is not enforced, hence the conclusion that torsion should vanish in the absence of a dilaton field can be avoided.
However, this would yield torsion with twist, which seems to imply a non-causal background [21]. Of course a more general choice of ℵ can resolve this issue altogether and yield finite and twistless torsion but we do not have any means to check this.
On a different note, one naturally wonders about the fate of the classical spacetime symmetries that we discussed in section II C 1. As discussed in section III B we can choose a measure that is invariant under the local Galilean boosts and rotations but in general it will break the U(1) central charge symmetry. In Appendix G we check the U(1) transformations of our beta functions. We find that, luckily, all the beta functions except β rs are invariant and in the absence of matter (hence in the absence of torsion) the transformation of the latter can be put in a form that is almost a total derivative plus a term that involves D tυ t . Our work can be improved and generalized in a number of ways. First, it is desirable to obtain the O(l 2 s ) contributions to the dilaton beta function. As mentioned above, this requires two-loop calculations on the worldsheet which can be done in the case of the bosonic string with relative ease but in our case there exists more than 20 contributions with different structures and this computation becomes a formidable task. Yet, it is straightforward and should be done in the near future. One may also consider a more general ansatz for the Kalb-Ramond field than we have taken in this paper. In particular, one can consider a generic, non-vanishing ℵ. However the beta function calculations will be modified drastically and the technology developed in this paper does not seem sufficient to calculate the beta functions in this case. Furthermore, it will be curious to compare our equations with the ones obtained from other effective approaches, such as the action principle proposed in [29], and the large c expansion of general relativity equations in [13]. Finally, it is very interesting to ask whether one can obtain Weyl invariant sub-critical TNC backgrounds with dimensionality less than 25 by searching for analogs of the linear-dilaton type geometry in the ordinary bosonic string case. In that case the slope of the linear dilaton cancels the O(l 0 s ) contribution to the dilaton beta function hence lifting the condition d = 26 and allowing for non-Lorentz invariant backgrounds with an arbitrary 2 < d < 26. Since we already gave up Lorentz invariance in the target spacetime, it is natural to ask if one can obtain subcritical TNC geometries with an analogous mechanism. To see if this is possible one will need the O(l 2 s ) contributions to the dilaton beta function.
y Vinculacion de Recursos Humanos de Alto Nivel.

Appendix A: Covariant expansion coefficients
The tensor coefficients {C mnrs , A smn , B rsm } from the covariant background expansion (50) are given by where we have used the following TNC identities The tensor coefficients {C mnrs ,Ā smn } from the covariant background expansion of the B-field are given by The relevant flat indices tensor coefficients {H, Z} coming from the B-field coupling are given by The gauge symmetries of the theory are with ω ± parametrizing local worldsheet Weyl/Lorentz transformations and ξ µ parametrizing worldsheet diffeomorphisms. Following the Faddeev-Popov procedure we can first compute the Faddeev-Popov determinant where {c, d ± , b * ,b * } are Faddeev-Popov ghosts and anti-ghosts and {a µ ,ā µ } are bosonic Lagrange multipliers enforcing the gauge conditionê. The remaining BRST symmetry of the theory is given by Integrating over {a,ā, d ± } imposes the constraints and the action simplifies into where we have omitted the hat on the vielbeins for simplicity. By considering the action (B2) (after gauge-fixing), it is now possible to define the ghost energy momentum one form as in (36) where we anticipated going to conformal gauge, i.e. the vielbeins are constant.
It is important to note that the anti-ghosts {b * ,b * } will not be neutral under local Weyl transformations, meaning that we will need to supplement the theory with the transforma- this implies the full condition for Weyl invariance is not (35) but rather whereτ c γ = τ c γ +τ c γFP is the total energy momentum one form and {B β ,B β } are the equations of motion for the anti-ghosts defined as In conformal gauge the ghost action takes the dimensionally extended form where we have defined {b ≡ b * − ,c ≡ c − , b ≡b * + , c ≡ c + } and we have rescaled the ghosts such that the normalizaion of the action is −1/2π. The non vanishing real space propagators are where ∆ 2 is an overall factor that will not play any role in our results. To find these propagators we used the identity ∂ 2 log |∆σ| 2 = 4π δ(∆σ) .

(B18)
Appendix C: Correlation functions in flat TNC background In this section wi will consider the dimensional extension to n dimensions of the more general free action S [0] 0 given in (53). We will in fact consider a more general case with ℵ m = Υτ m with constant Υ. In section (III E) we set Υ = 0. The action can be written in the conformal gauge as where Υ ± = 1 ± Υ, and we made the change of variables using e α ± ∂ α = 2∂ ± . The propagator for this free theory can now be computed as For the computation of (F4) shown in appendix F we will need the Weyl variation of two and four point functions. We will work in the context of dimensional regularization and consequently we can note that only logarithmically divergent correlators will contribute.
The relevant divergent two and four point functions for n = 2 + are where µ is a reference mass scale and we used standard dimensional regularization formulae to compute integrals over momenta [30]. The Weyl variation of these correlation functions is To compute these correlators we will need the following real space propagators (that can be read from (C3)): where ∆ 2 is an unimportant overall factor which was introduced after (B17). The contribution to the two point function from the constraints can then be computed to be where σ is the d'Alembertian on the worldsheet, σ = γ αβ ∂ α ∂ β . To rewrite this in a useful way we need the equations of motion for the classical fields. These are found by varying the Lagrangian (18): where We now multiply the first equation by 1 2 h pr , the second equation by e β + ∂ β and the third one by e β − ∂ β to find − σ X r = (Γ r mn +υ r ∂ m τ n ) ∂ α X m ∂ β X n γ αβ − (τ m + ℵ m ) X m = e α + e β − (∂ m τ n + ∂ m ℵ n ) ∂ α X m ∂ β X n − σ η (E8) (τ m − ℵ m ) X m = e α + e β − (∂ n τ m − ∂ n ℵ m ) ∂ α X m ∂ β X n + σ η (E9) where we have also used (E5) to simplify (E7). By adding and subtracting (E8) and (E9) we find Substituting (E10) in (E7) we finally have − σ X r = Γ r mn γ αβ +υ r ∂ m ℵ n αβ − 1 2 h rp H pmn αβ ∂ α X m ∂ β X n Now that we have an expression for σ X r in terms of ∂ α X r we can rewrite (E1) as from which one can easily read the dilaton contributions to the beta functions (F4).
where we have also included the contribution coming from ℵ m and the dilaton φ in the special (tractable) case ℵ m = Υτ m . Notice in particular that when ℵ m = ±τ m we have β = 0 and torsion is not forced to be twistless anymore.
The only way to make the measure invariant is to set D s σ = τ s F (X), since then one finds that vanishes by choosing f (Φ) = Φ.
For zero torsion this particular choice of gauge transformation leaves invariant the beta functions too as can be seen by substituting in (G2) and noticing that D 2 σ = D m h mn τ n F (X) = 0. Hence, at least for the zero torsion case, the breaking of the (full) U (1) symmetry at the quantum level can be traced to the impossibility of finding a measure that is invariant under this symmetry.