Comments on Weyl invariance of string theories in generalized supergravity backgrounds

We revisit Weyl invariance of string theories in generalized supergravity backgrounds. In the previous work arXiv:1703.09567, a possible counterterm was constructed, but it seems to be a point of controversy in some literatures whether it is non-local or not. To settle down this issue, we show that the counterterm is definitely local and exactly cancels out the one-loop trace anomaly in generalized supergravity backgrounds.


Introduction
A great progress in the recent study of String Theory is the derivation of the generalized supergravity equations of motion (GSE) [1,2] 1 from the kappa-symmetry constraints in the Green-Schwarz (GS) formulation of superstring theories [2].It has been well known that the usual supergravity equations of motion are solutions to the kappa-symmetry constraints [6,7], but the discovery of this new solution may indicate that there could be more veiled solutions.
In this note, we are concerned with string theory defined on generalized supergravity backgrounds (i.e.solutions of GSE).As a remarkable characteristic of GSE, a non-dynamical vector field I is contained.According to the kappa-symmetry constraints, this vector field should be a Killing vector, and this Killing condition plays a crucial role in our discussion.It would be pedagogical to note here that this Killing condition was not taken into account in 1 Historically speaking, the original discovery of the generalized type IIB supergravity was in the study of Yang-Baxter deformations of the AdS 5 ×S 5 superstring [3,4], though the bosonic part comes from a much older work by Hull and Townsend [5].
old literatures [5,8], where a prototype of GSE was derived from the one-loop finiteness (or the scale invariance) of string theory.In addition, this extra vector field can be identified with the trace of non-geometric Q-flux, and many solutions of GSE can be regarded as T -folds [9].
There is an issue with the consistency of string theories in generalized supergravity backgrounds.As a matter of course, at classical level, there is no problem.Thanks to the work [2], the kappa symmetry is ensured in generalized supergravity backgrounds and the GS formulation is consistently defined.The issue arises at quantum level.Indeed, the Weyl anomaly appears in string theory on generalized supergravity backgrounds [1,5].Recently in [10], Weyl invariance of bosonic string theories on generalized supergravity backgrounds was discussed and a possible counterterm was constructed as2 When I i = 0 , the usual coupling to dilaton Φ (often called the Fradkin-Tseytlin term) [11] is reproduced.In this sense, this is a generalization of the Fradkin-Tseytlin term.Compared to the sigma model action, the counterterm (1.1) is higher order in α ′ , and it should be regarded as a quantum correction.It is also noted that the Killing vector I , which appears in the equations of motion of the generalized supergravities, does not appear in the classical string sigma model action, but enters firstly as a quantum correction at stringy level.
The counterterm (1.1) depends on the dual coordinate Ỹi .When we compute its contribution to the trace of the energy-momentum tensor T a a , we need to use the equations of motion of the double sigma model [19], These equations of motion imply that Ỹi is a non-local function of X m and one may suspect that the counterterm (1.1) is non-local as well.However, as we show in this note, we can construct a similar local counterterm by considering that the two-dimensional Ricci scalar R (2) is a total derivative3 and I is a Killing vector.Hence it is not necessary to care about the apparent non-locality of the counterterm (1.1).This is the main claim in this note.
This note is organized as follows.Section 2 is the basics on Weyl invariance of bosonic strings in general backgrounds.In Section 3, we consider bosonic strings in generalized supergravity backgrounds.In that case, the Weyl invariance appears to be broken at the one-loop level, but we find a local counterterm to cancel out the trace anomaly.In Section 4, we explain the relation between the counterterm found in Section 3 and the counterterm (1.1) constructed in the double sigma model [10].We also explain why a generalized supergravity background always has a linear coordinate dependence in the dilaton.The counterterm found in Section 4 includes a vector density α a .The explicit forms of α a are provided in Section 5 from several approaches.Section 6 is devoted to conclusion and discussion.In Appendix A, we explain that GSE can be regarded as a (formal) T -dual of the nine-dimensional gauged supergravity.Appendix B contains several known generalized supergravity backgrounds that were obtained through non-Abelian T -duality.There, we concretely show that these solutions are T -dual to linear dilaton backgrounds.We also show that the linear dilaton can be removed by performing a non-linear field redefinition (see [20] for similar examples).

The basics on Weyl invariance of bosonic string
Let us first recall the basics on Weyl invariance of bosonic string theory in D = 26 dimensions.
We shall begin with the conventional string sigma model, where ε 01 = 1/ √ −γ .Then the Weyl anomaly of this system takes the form, Here, the β-functions at the one-loop level have been computed (for example in [5]) as where D m and R mn are the covariant derivative and the Ricci tensor associated with the spacetime metric g mn and For the Weyl invariance of the worldsheet theory, it is not necessary to require β g mn = β B mn = 0 .As long as they take the form the Weyl anomaly has a simple form under the equations of motion where D a is the covariant derivative associated with γ ab and e.o.m.
∼ represents the equality up to the equations of motion.This can be canceled out by adding a counterterm, the so-called Fradkin-Tseytlin term [11], to the original action (2.1).
Therefore, as long as the target space satisfies the equations (2.4), namely the supergravity equations of motion, the Weyl invariance is ensured.As discussed in [21], equations (2.7) imply that is constant, and by choosing β Φ = 0 , we obtain the usual dilaton equation of motion.
The main observation of this note is that the requirement (2.4) is a sufficient condition for the Weyl invariance but is not necessary.

Local counterterm for generalized supergravity
Let us consider a milder requirement, where I m and Z m are certain vector fields in the target space, which are functions of X m (σ) .
When the β-functions have the form (3.1), the Weyl anomaly (2.2) becomes Then, there is a rigid scale invariance [5], but it has been believed that the Weyl invariance could be broken because the counterterm (2.6) cannot cancel out the anomaly (3.2).However, we will construct a modified local counterterm such that (2.6) vanishes on-shell.
Recalling that the two-dimensional Einstein-Hilbert action is a total derivative, we define a vector density α a that transforms as4 under diffeomorphisms on the world-sheet.We then introduce the counterterm as Note that this reduces to the Fradkin-Tseytlin term (2.6) when where we supposed the world-sheet has no boundary.Assuming that Z m and I m are independent of γ ab , if we vary the counterterm with respect to γ ab , we obtain 4π √ −γ δS Here, suggested by the identity in two dimensions, we have used the variation where ǫ 01 = +1 and ϕ ab is a symmetric tensor made of the fundamental fields and their derivatives.Then, the contribution of the counterterm (3.5) to the Weyl anomaly becomes In fact, the divergence in the last term vanishes by using the on-shell conservation law of a Noether current (see Section 4), and we obtain Therefore, the anomaly (3.2) completely cancels.
Actually, the requirement (3.1) was derived as the condition for the one-loop finiteness of string sigma model [5].Now, we have proven that the Weyl symmetry can also be preserved upon introducing the above counterterm, so it is reasonable to expect that string theory can be consistently defined with the relaxed condition (3.1).In the following, we explain the condition (3.1) in terms of supergravity.

Generalized supergravity equations of motion
From (2.3) and (3.1), we can express the condition for the Weyl invariance as modified supergravity equations of motion, In fact, these are the generalized supergravity equations of motion for g mn and B mn proposed originally in [1].Such equations of motion were later derived in [2] from the requirement for the κ-invariance of the Green-Schwarz type IIB superstring theory.There, the vector fields are required to satisfy the following conditions: See [1,2,10] for the deformations of equations of motion for the Ramond-Ramond fields.In particular, when Z m = ∂ m Φ and I m = 0 , these reduce to the conventional supergravity equations of motion.
In general, Z m can be parameterized as Under the condition (3.14), we can choose a particular gauge where Ĩm vanishes [1,10].Therefore, in the generalized supergravity, the deformation is characterized only by the Killing vector I m .It is also noted that due to the presence of a Killing vector, solutions of the GSE are effectively nine dimensional.
In earlier works, many solutions of GSE have been obtained from the q-deformation [22], homogeneous Yang-Baxter deformations [9,[23][24][25], and non-Abelian T -duality [9,15,26] (see also [13])., despite there was not guarantee that these are string backgrounds.However, the cancellation of the Weyl anomaly that we provide here is an important step towards that direction 5 .
In the solutions of DFT, by using adapted coordinates where the Killing vector I m is constant, we find that the dilaton has a linear dependence on the dual coordinate xm [10].Moreover, if we perform a formal T -duality 6 along the I m -direction, an arbitrary solution of GSE is mapped to a solution of the conventional supergravity that has a linear coordinate dependence in the dilaton [1,10,32] (see Appendix B for examples).In the next section, we sketch the origin of the linear dilaton by introducing the Noether current associated with the Killing vector I m .

Linear dilaton in generalized supergravity backgrounds
In this section, we shall discuss a relation between the generalized supergravities and linear dilatons.These are closely related to each other intrinsically as we show below.
An arbitrary solution to the GSE admits a Killing vector I m by the definition of GSE, Due to the existence of the Killing vector, the string sigma model has a conserved current associated with the global symmetry where ǫ is an infinitesimal constant.Under an infinitesimal variation, δX m = ǫ I m , we obtain the (on-shell conserved) Noether current J a , Then, by recalling the parameterization (3.16), our counterterm (3.5) can be written as From the conservation law, J a can be represented by using a certain function Z(σ) as Then, the counterterm (4.4) can be further rewritten as Now, let us choose a particular gauge Ĩm = 0 .In this case, by comparing the relation

Constructions of local α a
In this section, we explain two ways to construct the vector density α a .Naively, from the defining relation, one might expect that α a can be expressed in terms of the metric γ ab .However, it is not the case as it was clearly discussed in [33,34].In order to construct α a in terms of the metric γ ab , we need to break the general covariance on the worldsheet.Indeed, the general solution obtained in [34] takes the form where λ is an arbitrary parameter that is coming from the ambiguity of α a α a → α a + ǫ ab ∂ b f . (5.3) If we consider its variation under an infinitesimal diffeomorphism δ v γ ab = £ v γ ab (with δ v λ = 0), we find it is not covariant, (5.4) Therefore, if α a is only written in terms of the metric and its derivatives, it will not be covariant.On the other hand, similar to the approach of [33], if we introduce a zweibein e āa on the worldsheet (ā and b are the flat indices), we find another expression up to the ambiguity (5.3) where ω āb c is the spin connection.In this case, despite α a is manifestly covariant under diffeomorphisms, it is not covariant under the local Lorentz symmetry.In the following, we explain two ways to provide covariant definitions of α a .

A construction using the Noether current
The first approach is based on the approach explained in Section II.B. of [34].In two dimensions, if there exists a normalized vector field n a (γ ab n a n b = ±1 ≡ σ), we can show that In string theory on generalized supergravity backgrounds, we have a natural vector field on the worldsheet, which is the Noether current J a in (4.3).Supposing J a is not a null vector on the worldsheet, we define the vector field n a as n a ≡ Then α a is defined as which is manifestly covariant and a local function of the fundamental fields.Moreover, by taking a variation of this α a in terms of γ ab , where the Noether current transforms as after a tedious computation, 7 we find the desired variation formula (3.9) with ϕ ab given by Therefore, this fully determines the variation of α a , for which the Weyl anomaly cancels out in generalized supergravity backgrounds.

A construction in the gauged sigma model
In the second approach, we introduce auxiliary fields to construct α a .For simplicity, here we choose a gauge Ĩm = 0 .
Let us consider the action of a gauged sigma model where The action reproduces the standard one (2.1) after integrating out the auxiliary field Z .In order to cancel out the one-loop Weyl anomaly, we add the following local term to S ′ : which is higher order in α ′ .The contribution to the trace of the energy-momentum tensor coming from S c is The equations of motion for A a and Z give where J a is the Noether current defined in (4.3).Since the field strength F ab vanishes to the leading order in α ′ , by using the local symmetry (5.11), we can find a gauge where the order O(α ′0 ) term vanishes (2) . (5.15) Here, A a is a quantity of order O(α ′0 ) .Then the trace (5.13) becomes This completely cancels the one-loop Weyl anomaly (3.2), which will be coming from S ′ .
After eliminating the auxiliary field Z , the action S ′ + S c becomes (5.17 As it is clear from (5.15), the gauge field A a plays the role of the desired α a via α a = −2 ǫ ab A b .
Then, we obtain and by neglecting the higher order term in α ′ , this is precisely the same as the standard sigma model action including our local counterterm S (I,Z) FT (3.5).
It is noted that the second line in the action (5.18) is the same as Eq.(5.13) of [16].There, it was obtained by rewriting the non-local piece of the effective action S non-local of [12] through the identifications of I m and Z m with some quantities in Yang-Baxter sigma model.In [12], the non-local action S non-local appeared in the process of non-Abelian T -duality, and it played an important role to show the tracelessness of T ab .However, according to the non-local nature of the effective action, by truncating the non-linear term by hand, it was concluded in [12] that the string model (called the B'-model) is scale invariant but not Weyl invariant.On the other hand, the action (5.18) or our local counterterm (3.5) with α a defined as (5.7) is local and free from the Weyl anomaly.

Conclusion and Discussion
In this note, we have constructed a local counterterm (3.5) that cancels out the Weyl anomaly of bosonic string theory defined in generalized supergravity backgrounds, without introducing a T -duality manifest formulation of string theory.This result clearly shows the Weyl invariance of string theory in generalized supergravity backgrounds.In order to claim the consistency of string theory in generalized supergravity backgrounds, it may be necessary to study some aspects of the associated CFT picture in more detail, but the first non-trivial test has been passed.Here, we have considered the case of bosonic string theory, but the same counterterm should work in the RNS superstring theory as well.
Our result indicates new possibilities of string theory in more general backgrounds.In fact, if we appropriately choose the parameters of the nine-dimensional gauged supergravity [35,36] and perform a formal T -duality along the ten-dimensional direction, we can obtain the GSE (see Appendix A).In DFT or its extension, the exceptional field theory, we can construct various deformed supergravities that are similar to GSE by performing the formal T -dualities and S-dualities [37].It is important to study the consistency of string theories defined on solutions of these deformed supergravities.A natural conjecture is that as long as the target space satisfies the equations of motion of the exceptional field theory, the string theory is consistently defined.We hope to come back on this interesting topic in our future researches.

B Examples of generalized supergravity backgrounds
In the appendix, we consider some known solutions to GSE that are obtained via non-Abelian T -dualities [9,15,26].We then show that the T -dualized background is a solution of supergravity that has a linear dilaton.We also find a combination of T -dualities and coordinate transformations that removes the linear dilaton.

B.1 Example 1
Let us consider the following background, which was studied in [9,26] ds where ds 2 M 6 is a six-dimensional flat metric.This is a solution of GSE.By performing a T -duality along the x-direction, we obtain a solution of the supergravity In fact, this is the original background before performing the non-Abelian T -duality (see (2) and (37) in [26]).Namely, the non-Abelian T -duality can be realized as a combination of Abelian T -dualities and coordinate transformations.

. 14 )
Using these conditions, equations of motion (3.13) lead to the following generalized dilaton equation of motion:R − 1 12 H mnp H mnp + 4 D m Z m − 4 (I m I m + Z m Z m ) = 0 .(3.15)Equations of motion (3.13) and (3.15) define the NS-NS sector of the generalized supergravity.

( 4 . 5 )
with the equations of motion of the double sigma model (1.2), we can identify Z with a combination of the dual coordinates I m Xm .Then, (4.6) is precisely the counterterm presented in[10], namely the Fradkin-Tseytlin term with the modified dilatonΦ * = Φ + I m Xm .(4.7)In this manner, thanks to the Killing property of I m , the difference between the standard Fradkin-Tseytlin term and our counterterm (4.4) can always be expressed as a linear dualcoordinate dependence in the dilaton.As we concretely show in Appendix B, by performing a formal T -duality in DFT along the Killing direction, this linear dual-coordinate dependence becomes a linear dependence on the physical coordinate X m .
and I ≡ I m ∂ m satisfies the Killing equations.This theory has a local symmetry,