Relating the theories

Replacing the structure coefficients F_ABC by covariant fluxesF_AB requires a computation of the Christoffel symbols, again. We follow [3]. It results in having to solve the frame field compatibility condition

∇_AE_B^I =D_AE_B^I +1

3F_BA^CE_C^I+E_A^KΓ_KJ^IE_B^J = 0, (4.141) and subsequently

Γ_IJ^K =−Ω_IJ^K+ Ω_{[IJ L]}η^LK = 1

3 −2Ω_IJ^K+ Ω^K_IJ + Ω_J^K_I

. (4.142)

The torsion of this generalized connection vanishes

T^I_{J K} = 2Γ_{[J K]}^I + Γ^I_{[J K]}= 0. (4.143)

Furthermore, the C-brackets (2.42),(3.67) of both theories are connected through the torsion by

ξ₁, ξ₂I C =

ξ₁, ξ₂I

DFT,C+T^I_{J K}ξ₁^Jξ₂^K. (4.144) Both theories are governed by the same gauge algebra, however the strong constraints (4.140) need to be exchanged. The same argumentation holds for the generalized Lie derivative as well

L_ξV^I = ξ, VI

C +1

2∇^I ξ_JV^J

= ξ, VI

DFT,C+1

2∂^I ξ_JV^J

=L_DFT,ξV^I, (4.145) as it can be expressed through the C-bracket. Therefore, any modification of the Christof-fel symbols does not change their transformation behavior (4.101) under 2D-diffeomor-phisms. Hence, the action and gauge transformations retain their 2D-diffeomorphism invariance. Unfortunately, the O(D, D) preserving constraint (4.120)

L_ξη^IJ = 0 =∂^Iξ^J+∂^Jξ^I, (4.146) partially violates the 2D-diffeomorphism symmetry. On top of that, the extended strong constraint of DFT_WZW and strong constraint of toroidal DFT for the generalized vielbein E_A^I should transform covariantly. This leads to the following auxiliary constraints

∆_ξ ∂_IE_A^J∂^If

=−E_A^K∂_K∂_Iξ^J∂^If = 0, (4.147)

∆_ξ ∂_IE_A^J∂^IE_B^K

=−E_A^L∂_L∂_Iξ^J∂^IE_B^K−∂_IE_A^JE_B^L∂_L∂^Iξ^K = 0, (4.148) and consequently require

∂_Iξ^J∂^If = 0, and ∂_Iξ^J∂^IE_A^K = 0 or ∂_Iξ^K = const. (4.149)

4. Generalized Metric Formulation of DFT on Group Manifolds

In this equation, the latter term allows for global O(D, D) rotations [3]. Except them, only the following transformations are allowed for the generalized vielbein

L_ξE_A^I =ξ^J∂_JE_A^I+E_A^J∂_Jξ^I =E_A^J

0 0

∂_[jξ˜_i] 0

, (4.150)

corresponding to B-Field gauge transformations

B_ij →B_ij +∂_[iξ_j], (4.151)

which we can express through generalized diffeomorphisms, as in the case of global O(D, D) rotations. This implies that the extended strong constraint (4.140) and the O(D, D) generalized background vielbein break the DFT_WZW 2D-diffeomorphism com-pletely.

Moreover, the newly introduced generalized connection (4.141) also has an affect on the background dilaton ¯d(4.18) in a non-trivial fashion. Correspondingly, the background dilaton is required to satisfy the compatibility condition for partial integration (3.11)

Γ_IJ^I = Ω^I_IJ =−2∂_Jd .¯ (4.152) In flat indices it takes on the form

F_A= Ω^A_AB+ 2D_Ad¯= 0. (4.153) The DFT_WZW backgrounds fulfill this relation by default, as they originate from a gener-alized Scherk-Schwarz ansatz which inherits (4.153) as a consistency requirement.

At this point, we want to recast the DFT_WZW action in the following way

S =S_{DF T} +S_∆, (4.154)

with S_{DF T} representing the original DFT action S_{DF T} =

Z

d^2DXe^−2d1

8H^KL∂_KH_IJ∂_LH^IJ − 1

2H^IJ∂_JH^KL∂_LH_IK

−2∂_Id ∂_JH^IJ + 4H^IJ∂_Id ∂_Jd

, (4.155)

and the auxiliary term

S_∆=−2 3

Z

d^2DXe^−2dH^IJK_IJ. (4.156) However, this separation of the action is only valid under application of the extended strong constraint (4.140), with K_IJ arising from the DFT field equations. (See [46] for more details). After performing the necessary projections (2.84), which are required due to the undetermined components of this tensor, it allows us to obtain the generalized

4.5. Relation to original DFT Ricci tensor R_IJ and the corresponding the equations of motion. For the choice of our Scherk-Schwarz background, we find

K_IJ = 1

4F_IKLF_{J M N} η^KMη^LN−H^KMH^LN

. (4.157)

The difference between all the metrics H^IJ, combining background and fluctuations, and H^IJ, describing the background only, should be kept in mind. If equation (4.154) holds, the remaining term

S−S_{DF T} −S_∆= Z

d^2DXe^−2d∆ (4.158)

must vanish. We now start to replace all covariant derivatives by partial derivatives and the according generalized connection (4.142). It yields

∆ =H^IJ

Ω_IKLΩ^KL_J−Ω^K_KIΩ^L_LJ +1

2Ω_KLIΩ^KL_J

(4.159)

−Ω_IJ^K∂_KH^IJ + 2Ω^K_KIH^IJ∂_Jd˜−Ω^K_KI∂_JH^IJ + 2H^IJΩ_IJ^K∂_Kd .˜

As a result of the strong constraint for background fields, the last term in the first line becomes zero. Furthermore, we perform a partial integration (3.10) and split the dilaton according to (4.18). We obtain

−Ω_IJ^K∂_KH^IJ =−2H^IJΩ_IJ^K∂_Kd˜+H^IJΩ_IJ^KΩ^L_LK+∂_KΩ_IJ^KH^IJ, (4.160)

−Ω^K_KI∂_JH^IJ =−2Ω^K_KIH^IJ∂_Jd˜+H^IJΩ^K_KIΩ^L_LJ+H^IJ∂_IΩ^K_KJ. (4.161) In order to get rid of the derivatives acting on ¯d, we applied equation (4.153) . Next, we utilize the newly introduced identities (4.160), (4.161). As an immediate consequence, ∆ reduces to

∆ = H^IJ

Ω_IKLΩ^KL_J+ Ω_IJ^KΩ^L_LK +∂_KΩ_IJ^K +∂_IΩ^K_KJ

. (4.162)

During the last step, we use the definition of the coefficients of anholonomy (4.100). A straight forward computation gives rise to

∂_KΩ_IJ^K+∂_IΩ^K_KJ =−Ω_IJ^KΩ^L_LK −Ω_IKLΩ^KL_J. (4.163) Finally, we arrive at the desired result

∆ = 0. (4.164)

Moreover, we are interested in evaluating K_IJ. Therefore, we switch back to flat indices and exploit the strict left/right segregation of the structure coefficients (4.4). We obtain F_ACEF_BDFη^CDη_EF =F_ACEF_BDFS^CDS^EF. (4.165)

4. Generalized Metric Formulation of DFT on Group Manifolds

Thus, we immediately find K_AB = 1

4F_ACEF_BDF

η^CDη^EF −S^CDS^EF

= 0. (4.166)

Right away, it follows K_IJ = 0, in curved indices, and the main result of this section

S =S_{DF T} , (4.167)

that the action of DFT_WZW reduces to the one of original DFT under the imposition of the extended strong constraint [3].

At last, we show that its possible to choose an arbitrary realization of the generalized vielbein E_A^I as long as the strong constraint is fulfilled. In this context, we argue why the O(D) × O(D) gauge fixing of section 4.5.1 is very convenient. Therefore, we show that S = S_{DF T} is invariant under local O(D) × O(D) transformations. Clearly, these transformations leave

δ_Λη^AB = Λ^A_Cη^CB+ Λ^B_Cη^AC = 0, and δ_ΛS^AB = Λ^A_CS^CB + Λ^B_CS^AC = 0 (4.168) invariant. A suitable way to show this requires us to compute the failure of K_AB to transform covariantly

∆_ΛK_AB = 0. (4.169)

With regards to this, it is useful to take the failure of the covariant fluxes [109] into account

∆_ΛF_ABC = 3D_[AΛ_BC]. (4.170)

From which we obtain

∆_ΛK_AB = 3

4 D_[AΛ_CE]F_BDF +D_[BΛ_DF]F_ACE

η^CDη^EF −S^CDS^EF

. (4.171)

Making use of the O(D, D) condition

S^ACΛ_CDS^DB = Λ^AB, (4.172)

analogous to equation (4.165), we acquire the wanted result

∆_ΛK_AB = 0. (4.173)

Thus, we can freely choose an arbitrary realization of the generalized backgroundE_A^I, as long as it satisfies the strong constraint. For instance, we could choose the bivector β^ij instead of the B-field B_ij, or a vielbein which neither lies in the left- nor right-moving Maurer-Cartan form.

Summarizing, we can view the entire computation executed in this subsection as kind of a generalization of the steps performed to find a background independent action [45]

4.5. Relation to original DFT of the cubic DFT action [1]. The proof of the background independence has been ac-complished by absorbing the constant part of the fluctuations _ij into a change of the background field E_ij, where the dilaton does not contribute in any way [45]. In our case, it is a very similar situation. The generalized metric (4.7) is split into a background field H^IJ and fluctuations h^IJ through

H^IJ =H^IJ +h^IJ, with h^IJ =^IJ +1

2^IKH_KL^LJ +. . . . (4.174) Although, in contrast to [45] we have to consider generalized dilaton contributions as well. Moreover, the background field H^IJ is generally not constant for arbitrary group manifolds. This indicates that we are not solely restricted to constant background fields.

Only for an underlying group manifold such as the torus we obtain a constant background.

Furthermore, the field equations must always hold for a consistent background. However, we are still able to reproduce the background independence known from original DFT [3, 45].

Therefore, this subsection shows that in order for DFT_WZW to be background inde-pendent, we must impose the extended strong constraint (4.140). It rules out all solutions beyond the SUGRA regime. Additionally, DFT_WZW might give insights into new non-geometric background and physics which are going beyond the SUGRA/DFT regime despite inheriting the same background independence as in original DFT. On top of that, DFT breaks DFT_WZW’s 2D-diffeomorphism invariance as the derivation in this subsection clearly shows.

5. Flux Formulation of DFT on Group

Im Dokument Advancements in double & exceptional field theory on group manifolds (Seite 89-95)