Comments on Weyl invariance of string theories
arXiv:1811.10600v1 [hep-th] 26 Nov 2018
in generalized supergravity backgrounds
Jos´e J. Fern´andez-Melgarejoa,∗ Jun-ichi Sakamotob,† Yuho Sakatanic,‡ and Kentaroh Yoshidab§
a
Departamento de F´ısica, Universidad de Murcia b
c
Department of Physics, Kyoto University
Department of Physics, Kyoto Prefectural University of Medicine
Abstract 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.
∗
E-mail E-mail ‡ E-mail § E-mail
†
address: address: address: address:
[email protected] [email protected] [email protected] [email protected]
Contents 1 Introduction
1
2 The basics on Weyl invariance of bosonic string
3
3 Local counterterm for generalized supergravity
4
3.1
Generalized supergravity equations of motion . . . . . . . . . . . . . . . . . .
6
4 Linear dilaton in generalized supergravity backgrounds
7
5 Constructions of local αa
8
5.1
A construction using the Noether current . . . . . . . . . . . . . . . . . . . . .
9
5.2
A construction in the gauged sigma model . . . . . . . . . . . . . . . . . . . .
10
6 Conclusion and Discussion
11
A GSE as a formal T -dual of a 9D gauged supergravity
12
B Examples of generalized supergravity backgrounds
14
1
B.1 Example 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
14
B.2 Example 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
15
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 AdS5 ×S5 superstring [3, 4], though the bosonic part comes from a much older work by Hull and Townsend [5].
1
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 Z √ 1 d2 σ −γ R(2) Φ∗ , SFT = 4π
Φ∗ ≡ Φ + I i Y˜i .
(1.1)
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. A point of controversy in some literatures [12–18] is the locality of this counterterm (1.1). The counterterm (1.1) depends on the dual coordinate Y˜i . 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], ∂a Y˜i = gin εb a ∂b X n + Bin ∂a X n .
(1.2)
These equations of motion imply that Y˜i 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 2
This counterterm was inspired from the generalized Buscher rule [1] and the embedding into double field
theory [10]. For the detail of the notation, see [10]. 3 We appreciate J. Maldacena for elucidating this point.
2
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 αa . The explicit forms of αa 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).
2
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, Z √ 1 ab ab 2 −γ g γ − B ε ∂a X m ∂b X n , S=− d σ mn mn 4πα′ √ where ε01 = 1/ −γ . Then the Weyl anomaly of this system takes the form, g B 2α′ hT a a i = βmn γ ab − βmn εab ∂a X m ∂b X n .
(2.1)
(2.2)
Here, the β-functions at the one-loop level have been computed (for example in [5]) as 1 g βmn = α′ Rmn − Hmpq Hn pq , 4
1 B βmn = α′ − D k Hkmn , 2
(2.3)
where Dm and Rmn are the covariant derivative and the Ricci tensor associated with the spacetime metric gmn and Hmnp ≡ 3 ∂[m Bnp] . For the Weyl invariance of the worldsheet g B theory, it is not necessary to require βmn = βmn = 0 . As long as they take the form g βmn = −2 α′ Dm ∂n Φ ,
B βmn = −α′ ∂k Φ H k mn ,
(2.4)
the Weyl anomaly has a simple form under the equations of motion 2πα′ δS hT a a i = −D a ∂a Φ + D m Φ √ −γ δX m
3
e.o.m.
∼
−D a ∂a Φ ,
(2.5)
e.o.m.
where Da is the covariant derivative associated with γab and ∼ 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], SFT
1 = 4π
Z
d2 σ
√
−γ R(2) Φ ,
(2.6)
to the original action (2.1). Therefore, as long as the target space satisfies the equations (2.4), namely the supergravity equations of motion, Rmn −
1 Hmpq Hn pq + 2 Dm ∂n Φ = 0 , 4
1 − D k Hkmn + ∂k Φ H k mn = 0 . 2
(2.7)
the Weyl invariance is ensured. As discussed in [21], equations (2.7) imply that β Φ ≡ R + 4 D m∂m Φ − 4 |∂Φ|2 −
1 Hmnp H mnp , 12
(2.8)
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.
3
Local counterterm for generalized supergravity
Let us consider a milder requirement, g βmn = −2 α′ D(m Zn) ,
B βmn = −α′ Z k Hkmn + 2 D[mIn] ,
(3.1)
where Im and Zm are certain vector fields in the target space, which are functions of X m (σ) . The condition (3.1) reduces to (2.4) when Zm = ∂m Φ and I m = 0 . When the β-functions have the form (3.1), the Weyl anomaly (2.2) becomes hT a a i
= e.o.m.
∼
2πα′ δS −Da (Zm γ ab − Im εab ) ∂b X m + Z m √ −γ δX m −Da (Zm γ ab − Im εab ) ∂b X m .
(3.2)
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, √
−γ R(2) = ∂a αa , 4
(3.3)
we define a vector density αa that transforms as4 δξ αa = £ξ αa = ξ b ∂b αa − αb ∂b ξ a + ∂b ξ b αa , under diffeomorphisms on the world-sheet. We then introduce the counterterm as Z 1 (I,Z) SFT = − d2 σ αa Zm ∂a X m − Im εa b ∂b X m . 4π
(3.4)
(3.5)
Note that this reduces to the Fradkin–Tseytlin term (2.6) when I m = 0 and Zm = ∂m Φ , Z Z √ 1 1 (0,dΦ) 2 a d σ α ∂a Φ = d2 σ −γ R(2) Φ , (3.6) SFT = − 4π 4π where we supposed the world-sheet has no boundary. Assuming that Zm and Im are independent of γab , if we vary the counterterm with respect to γ ab , we obtain (I,Z) 4π δSFT √ = −D(a| Zm ∂|b) X m − Im ε|b) c ∂c X m + γab D c Zm ∂c X m − Im εc d ∂d X m ab −γ δγ αc Im εc(a ∂b) X m − 12 γab εc d ∂d X m +√ (3.7) −γ − ϕab Dc (Im γ cd − Zm εcd ) ∂d X m .
Here, suggested by the identity in two dimensions, δ
√
√ −γ R(2) = ∂c −γ γ ca D b δγab − γ ab D c δγab ,
(3.8)
we have used the variation δαc =
√
−γ γ ca D b δγab − γ ab D c δγab + ǫcd ∂d (ϕab δγab ) ,
(3.9)
where ǫ01 = +1 and ϕab 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 (I,Z)
hT iFT
4π ab δSFT =√ γ −γ δγ ab = Da Zm γ ab − Im εab ∂b X m − ϕa a Dc (Im γ cd − Zm εcd) ∂d X m .
(3.10)
In fact, the divergence in the last term vanishes e.o.m. Dc (Im γ cd − Zm εcd ) ∂d X m = Dc J c ∼ 0 ,
(3.11)
by using the on-shell conservation law of a Noether current (see Section 4), and we obtain hT iFT 4
e.o.m.
∼
Da Zm γ ab − Im εab ∂b X m .
Although the explicit form of αa is not important here, it is given in Section 5.
5
(3.12)
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.
3.1
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, Rmn −
1 Hmpq Hn pq + 2 D(m Zn) = 0 , 4
1 − D k Hkmn + Z k Hkmn + 2 D[m In] = 0 . 2
(3.13)
In fact, these are the generalized supergravity equations of motion for gmn and Bmn 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: £I gmn = 0 ,
I p Hpmn + 2 ∂[m Zn] = 0 ,
£I Φ = 0 ,
Zm I m = 0 .
(3.14)
Using these conditions, equations of motion (3.13) lead to the following generalized dilaton equation of motion: R−
1 Hmnp H mnp + 4 Dm Z m − 4 (I m Im + Z m Zm ) = 0 . 12
(3.15)
Equations of motion (3.13) and (3.15) define the NS–NS sector of the generalized supergravity. See [1, 2, 10] for the deformations of equations of motion for the Ramond–Ramond fields. In particular, when Zm = ∂m Φ and I m = 0 , these reduce to the conventional supergravity equations of motion. In general, Zm can be parameterized as Zm = ∂m Φ + I n Bnm + I˜m .
(3.16)
Under the condition (3.14), we can choose a particular gauge where I˜m 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. 6
In earlier works, many solutions of GSE have been obtained from the q-deformation [22], homogeneous Yang–Baxter deformations [9, 23–25], 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 direction5 . As discussed in [10, 27], we can regard solutions of GSE as solutions of the double field theory (DFT) [28–31], which is a manifestly T -duality covariant formulation of supergravity. 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 x˜m [10]. Moreover, if we perform a formal T -duality6 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 .
4
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, £I gmn = 0 ,
£I Bmn + ∂m I˜n − ∂n I˜m = 0 ,
£I Φ = 0 .
(4.1)
Due to the existence of the Killing vector, the string sigma model has a conserved current associated with the global symmetry Xm → Xm + ǫ Im ,
(4.2)
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 ,
5
J a ≡ I m gmn γ ab − Bmn εab − I˜n εab ∂b X n ,
Da J a
e.o.m.
∼
0.
(4.3)
The solution obtained from q-deformation includes an imaginary Ramond–Ramond field, and may not be
a string background. 6 A formal T -duality means the factorized T -duality along a non-isometry direction xz , which maps the coordinate xz into the dual coordinate x ˜z . Such transformation is a symmetry of the equations of motion of DFT.
7
Then, by recalling the parameterization (3.16), our counterterm (3.5) can be written as Z √ 1 (I,Z) d2 σ −γ R(2) Φ + εab αa J b . (4.4) SFT = 4π ˜ From the conservation law, J a can be represented by using a certain function Z(σ) as Ja
e.o.m.
∼
εba ∂b Z˜ .
Then, the counterterm (4.4) can be further rewritten as Z √ (I,Z) e.o.m. 1 ˜ . SFT ∼ d2 σ −γ R(2) (Φ + Z) 4π
(4.5)
(4.6)
Now, let us choose a particular gauge I˜m = 0 . In this case, by comparing the relation (4.5) with the equations of motion of the double sigma model (1.2), we can identify Z˜ with ˜ m . Then, (4.6) is precisely the counterterm a combination of the dual coordinates I m X presented in [10], namely the Fradkin–Tseytlin term with the modified dilaton ˜m . Φ∗ = Φ + I m X
(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 .
5
Constructions of local αa
In this section, we explain two ways to construct the vector density αa . Naively, from the defining relation, √
−γ R(2) = ∂a αa ,
(5.1)
one might expect that αa can be expressed in terms of the metric γab . However, it is not the case as it was clearly discussed in [33, 34]. In order to construct αa in terms of the metric γab , we need to break the general covariance on the worldsheet. Indeed, the general solution obtained in [34] takes the form i γ01 γ01 1 h 0 −λ ∂1 γ00 + (1 − λ) ∂1 γ11 − 2 (1 − λ) ∂1 γ01 + ∂0 γ11 , α =√ −γ γ00 γ11 i γ01 1 h γ01 1 −(1 − λ) α =√ ∂0 γ11 + λ ∂0 γ00 − 2 λ ∂0 γ01 + ∂1 γ00 , −γ γ11 γ00 8
(5.2)
where λ is an arbitrary parameter that is coming from the ambiguity of αa αa → αa + ǫab ∂b f .
(5.3)
If we consider its variation under an infinitesimal diffeomorphism δv γab = £v γab (with δv λ = 0), we find it is not covariant, δv αa 6= £v αa .
(5.4)
Therefore, if αa 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 ea¯ a on the worldsheet (¯a and ¯b are the flat indices), we find another expression up to the ambiguity (5.3) √ ¯ αa = −2 −γ ea¯ a ω¯b b¯a ,
(5.5)
¯
where ωa¯ b¯c is the spin connection. In this case, despite αa 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 αa .
5.1
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 na (γab na nb = ±1 ≡ σ), we can show that
√
−γ R(2) = 2 σ ∂a
√
−γ nb Db na − na Db nb
.
(5.6)
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 na as na ≡ √ αa ≡ 2 σ
√
Ja σ γcd J c J d
. Then αa is defined as
−γ nb Db na − na Db nb ,
(5.7)
which is manifestly covariant and a local function of the fundamental fields. Moreover, by taking a variation of this αa in terms of γab , where the Noether current transforms as √ √ δ( −γ J a ) = δ( −γ γ ab ) ∂b X m Im ,
9
(5.8)
after a tedious computation,7 we find the desired variation formula (3.9) with ϕab given by 2 b) (a c m d ab c (a b) (5.9) ε (c δd) D X Im n . ϕ = σ n εc n + p σ γgh J g J h
Therefore, this fully determines the variation of αa , for which the Weyl anomaly cancels out in generalized supergravity backgrounds.
5.2
A construction in the gauged sigma model
In the second approach, we introduce auxiliary fields to construct αa . For simplicity, here we choose a gauge I˜m = 0 . Let us consider the action of a gauged sigma model Z √ 1 ′ S =− d2 σ −γ gmn γ ab − Bmn εab Da X m Db X n − Z˜ εab Fab , ′ 4πα
(5.10)
where Da X m ≡ ∂a X m − I m Aa , Fab ≡ ∂a Ab − ∂b Aa , and I ≡ I m ∂m satisfies the Killing equations. This theory has a local symmetry, X m (σ) → X m (σ) + I m v(σ) ,
Aa (σ) → Aa (σ) + ∂a v(σ) .
(5.11)
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 ′ : Z √ 1 ˜ , d2 σ −γ R(2) (Φ + Z) Sc ≡ (5.12) 4π which is higher order in α′ . The contribution to the trace of the energy-momentum tensor coming from Sc is 4π ab δSc e.o.m. a ˜ . γ ∼ D (∂a Φ + ∂a Z) hT ic = √ −γ δγ ab
(5.13)
The equations of motion for Aa and Z˜ give ∂a Z˜ = εb a Jb − |I|2 εb a Ab ,
ǫab Fab = α′
√
−γ R(2) ,
(5.14)
where Ja is the Noether current defined in (4.3). Since the field strength Fab 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 Aa = 0 + α′ Aa , 7
√ ǫab (∂a Ab − ∂b Aa ) = − −γ R(2) .
We repeatedly use the identity 2 A···[a B b]··· = −εab εcd A···[c B d]··· satisfied in two dimensions.
10
(5.15)
Here, Aa is a quantity of order O(α′0 ) . Then the trace (5.13) becomes e.o.m. hT ic ∼ D a ∂a Φ + εb a Jb + O(α′ ) = Da Zm γ ab − Im εab ∂b X m + O(α′ ) .
(5.16)
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 ′ + Sc becomes Z √ 1 ′ ab ab 2 S + Sc = − −γ g γ − B ε ∂a X m ∂b X n d σ mn mn ′ 4πα Z √ 1 + d2 σ −γ R(2) Φ + ǫab (−2 ǫac Ac ) J b − α′ |I|2 γ ab Aa Ab . 4π
(5.17)
As it is clear from (5.15), the gauge field Aa plays the role of the desired αa via αa = −2 ǫab Ab .
Then, we obtain
Z √ 1 S + Sc = − d2 σ −γ gmn γ ab − Bmn εab ∂a X m ∂b X n ′ 4πα Z √ 1 |I|2 2 ab (2) a b ′ d σ −γ R Φ + ǫab α J + α √ 2 γ αa αb , + 4π 4 −γ ′
(5.18)
and by neglecting the higher order term in α′ , this is precisely the same as the standard sigma (I,Z)
model action including our local counterterm SFT
(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 Snon-local of [12] through the identifications of Im and Zm with some quantities in Yang–Baxter sigma model. In [12], the non-local action Snon-local appeared in the process of non-Abelian T -duality, and it played an important role to show the tracelessness of Tab . 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 αa defined as (5.7) is local and free from the Weyl anomaly.
6
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 11
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.
Acknowledgments K.Y. is very grateful to S. Iso, J. Maldacena, J.-H. Park, P. Townsend, A. Tseytlin and K. Zarembo for valuable comments and discussions. Discussions during “11th Taiwan String Workshop,” the workshop YITP-T-18-04 “New Frontiers in String Theory 2018” and “The 5th Conference of the Polish Society on Relativity” were useful to complete this work. J.J.F.M. acknowledges financial support of Fundaci´on S´eneca/Universidad de Murcia (Programa Saavedra Fajardo). The work of J.S. is supported by the Japan Society for the Promotion of Science (JSPS). The work of Y.S. is supported by JSPS Grant-in-Aids for Scientific Research (C) 18K13540 and (B) 18H01214. The work of K.Y. is supported by the Supporting Program for Interaction based Initiative Team Studies (SPIRITS) from Kyoto University and by a JSPS Grant-in-Aid for Scientific Research (B) No. 18H01214. This work is also supported in part by the JSPS Japan-Russia Research Cooperative Program.
A
GSE as a formal T -dual of a 9D gauged supergravity
In the appendix, we show that, by perfoming a formal T -duality, GSE are equivalent to the equations of motion of a nine-dimensional gauged supergravity studied in [35, 36]. For convenience, here we choose a gauge in which the Killing vector has the form I = m ∂y (y ≡ x9 , m: constant) and I˜m = 0 . As discussed in [10, 27], GSE can be derived from the equations of motion of DFT by
12
making an ansatz for the bosonic fields, gmn = gmn (xµ ) ,
Bmn = Bmn (xµ ) ,
Cˆp = Cˆp (xµ ) .
Φ = φ(xµ ) + m y˜ ,
(A.1)
where µ = 0, . . . , 8 , y˜ is the dual coordinate associated with y , and Cˆp is the Ramond–Ramond
potential (see [10] for our convention). If we perform a formal T -duality along the y-direction, this becomes ′ ′ gmn = gmn (xµ ) ,
′ ′ Bmn = Bmn (xµ ) ,
Φ′ = φ′ (xµ ) + m y ,
Cˆp′ = Cˆp′ (xµ ) .
(A.2)
Removing the prime, we obtain m
Gmn = e− 2 y Gmn (xµ ) ,
Bmn = Bmn (xµ ) ,
Φ = φ(xµ ) + m y ,
ˆ p (xµ ) , (A.3) Cˆp = e−m y C Φ
where we have introduced the Einstein-frame metric Gmn = e− 2 gmn and the standard φ ˆ p ≡ e−φ Cˆp . Ramond–Ramond potential Cˆp ≡ e−Φ Cˆp with Gmn ≡ e− 2 gmn and C The ansatz (A.3) is precisely the one used in [35] to obtain a nine-dimensional gauged supergravity from 10D effective theories. Indeed, in the type IIA case given in (C.9) of [35], by choosing mIIA = −
2m , 9
m4 =
4m , 3
(A.4)
(A.3) is recovered. Similarly, in the type IIB case given in (C.14) of [35], (A.3) is recovered by choosing mIIB = −
m , 4
m1 = −m ,
m2 = m3 = 0 ,
α=−
m . 2
(A.5)
Therefore, both the type IIA/IIB GSE are related to the well-known nine-dimensional gauged supergravity through a formal T -duality in DFT. As shown in Figure 1 of [35], when a mass parameter mIIA or mIIB (which corresponds to a scaling symmetry, called trombone symmetry) is turned on, the gauged supergravity does not have the action which derive the equations of motion. This is consistent with the absence of the supergravity action for GSE [1]. For the gauged supergravity where the trombone symmetry is gauged, higher-derivative corrections are not known in the literature. Therefore, it is interesting to study the higher-derivative corrections for GSE by computing the β-functions at the one-loop level. For a general nine-dimensional supergravity without choosing the above parameters, we can still perform a formal T -duality and find a generalization of GSE, where the Killing vector I m enters the equations of motion in a more complicated manner. It is also interesting to study string theory in solutions of such generalized gauged supergravities. 13
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] (t4 + z 2 ) dy 2 − 2 y z dy dz + (t4 + y 2) dz 2 + t4 dx2 ds = −dt + + ds2M6 , 2 4 2 2 t (t + y + z ) (y dy + z dz) ∧ dx 1 1 , I = −2 ∂x , B2 = , Φ = ln 2 4 t4 + y 2 + z 2 2 t (t + y 2 + z 2 ) 2
2
(B.1)
where ds2M6 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 (t4 + y 2 + z 2 ) dx2 + 2 (y dy + z dz) dx + dy 2 + dz 2 + ds2M6 , t2 Φ = −2 (ln t + x) ,
ds2 = −dt2 + B2 = 0 ,
(B.2)
which has a linear x-dependence in the dilaton. By further performing a coordinate transformation t≡T,
x ≡ − ln X ,
y≡Y X,
z ≡ZX,
(B.3)
we obtain a simple solution ds2 = −dT 2 +
T2 X2 2 dX + dY 2 + dZ 2 + ds2M6 , 2 2 X T
B2 = 0 ,
Φ=
1 hX4 i ln 4 . 2 T
(B.4)
Finally, performing T -dualities along the Y - and Z-directions, we obtain a purely gravitational background ds2 = −dT 2 + T 2
dX 2 + dY 2 + dZ 2 X2
+ ds2M6 ,
B2 = 0 ,
Φ = 0.
(B.5)
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
B.2
Example 2
The second example is the solution of GSE obtained in [15] t2 2 2 dx + dy + ds2M7 , t4 + y 2 y 1 B2 = 4 dx ∧ dy , Φ = − ln t4 + y 2 , 2 t +y 2
ds2 = −dt2 +
(B.6) I = ∂x ,
where ds2M7 is a seven-dimensional flat metric. Again by performing a T -duality along the x-direction, we obtain a linear-dilaton solution (t4 + y 2 ) dx2 − 2 y dx dy + dy 2 + ds2M7 , t2 Φ = −(ln t − x) .
ds2 = −dt2 + B2 = 0 ,
(B.7)
By further performing a coordinate transformation t≡T ,
x ≡ ln X ,
y≡Y X,
(B.8)
we obtain ds2 = −dT 2 +
T2 X2 2 dX + dY 2 + ds2M7 , X2 T2
B2 = 0 ,
Φ=
Finally, we perform a T -duality along the Y -direction and obtain dX 2 + dY 2 + ds2M6 , B2 = 0 , ds2 = −dT 2 + T 2 X2
1 hX2 i ln 2 . 2 T
(B.9)
Φ = 0.
(B.10)
This is again the original background before performing the non-Abelian T -duality.
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