6. Generalized Scherk-Schwarz Compactifications in DFT on Group Manifolds 93
6.3. DFT on group manifolds
6. Generalized Scherk-Schwarz Compactifications in DFT on Group Manifolds
is trivial in these directions. We neglected the global factor here. As expected before, the action (6.31) describes a bosonic subsector of a half-maximal, electrically gauged supergravity [4]. It is identical to the approach demonstrated in [116].
All of the previous derivations in this subsection only took the properties (6.19)-(6.21) of the twistUIJ into account. Nevertheless, in general it is unclear whether there exists a twist with these exact properties for every embedding tensor solution. In original DFT, there does not exist a systematic way to construct these twists and as a result one is left with guessing solutions of partial differential equations (6.19) which are constrained to be elements of O(n, n) as well. This task proves to be highly difficult. Some of these solutions have been discussed in [65, 82] and more recently in the context of Exceptional Field Theory (EFT) [108]. The problem regarding the twist is one of the major differences between geometric Scherk-Schwarz compactifications [119, 120], which have been known for several years in the context of supergravity compactifications, and their generaliza-tion in DFT. For the former compactificageneraliza-tions, there exists a straightforward, systematic approach to construct their twists. One either uses the left or right invariant Maurer-Cartan form on the group manifold the compactification is considered on. However, for original DFT this systemic procedure is not applicable anymore, as it requires a geometry governed by ordinary diffeomorphisms as opposed to the O(n, n) preserving generalized diffeomorphisms appearing in DFT. In the remainder of this chapter, we are going to show that DFTWZW is able to cure this problem. It can be understood from the fact that all background fields transform covariantly under 2D-diffeomorphisms and we therefore recover the common notion of geometry. As a consequence, subsection 6.4 shows that in DFTWZW one can construct a twist/background vielbein by either using a left or right invariant Maurer-Cartan form [4].
6.3. DFT on group manifolds where only two different indices structures appear. The reason for this lies in the fact that DFTWZW possesses an additional background vielbein. In our context, the external indices ˆa, a and µ run from 0 toD−n−1 and their internal counterparts ˆA, A and M parameterize a 2n-dimensional, doubled space. As a consequence, we obtain the following three different versions of the η-metric
ηˆ˜
ABˆ˜ =
0 δˆaˆb 0 δˆaˆb 0 0 0 0 ηAˆBˆ
ηA˜B˜ =
0 δab 0 δab 0 0 0 0 ηAB
ηM˜N˜ =
0 δµν 0 δµν 0 0 0 0 ηM N
, (6.34) we are employing to lower the indices defined in (6.33). Furthermore, we work with the flat, background generalized metric
SAˆ˜Bˆ˜ =
ηˆaˆb 0 0 0 ηˆaˆb 0 0 0 SAˆBˆ
and its inverse SAˆ˜Bˆ˜ =
ηaˆˆb 0 0 0 ηaˆˆb 0 0 0 SAˆBˆ
. (6.35) In the next step, we need to specify the Scherk-Schwarz ansatz of the composite generalized vielbein
Eˆ˜
A M˜
= ˜Eˆ˜
A B˜
(X)EB˜ M˜
(Y). (6.36)
Here, we use the ansatz that the fluctuation part only depends on the external coordinates X, whereas the background part only depends upon the internal ones Y. Comparing our ansatz with the one in [41, 105, 113, 116], we observe that the background generalized vielbein EB˜
M˜
plays the role of the twist UNˆMˆ. In contrast to the twist appearing in original DFT, our background vielbein is not restricted to lie in O(D, D). Therefore, our framework allows to solve the problem of constructing an appropriate twist: We always possess a straight forward procedure to construct EB˜
M˜
by using the left invariant Maurer Cartan form on a group manifold [4]. One possible example to apply this technique would be the S3 with H-flux presented in [82].
Subsequently, we adapt the generalized Kaluza-Klein ansatz [41, 105, 113] for the fluctuation vielbein ˜EAˆ˜
B˜
and its associated index structure. It yields
E˜Aˆ˜ B˜
(X) =
ebˆa 0 0
−eaˆcCbc eˆab −eˆacAbBc EbAˆ
C
AbCb 0 EbAˆ B
with Cab =Bab+ 1
2AbDaAbDb. (6.37) During this ansatz, Bab represents the two-form field arising in the effective theory while
HbCD =EbAˆ
CSAˆBˆEbBˆ
D (6.38)
6. Generalized Scherk-Schwarz Compactifications in DFT on Group Manifolds
denotes the n2 independent scalar fields forming the moduli of the internal space. In the same fashion as for the twist, the background vielbein has only non-trivial components in the internal space. This gives rise to
EB˜ M˜
(Y) =
δbµ 0 0 0 δbµ 0 0 0 EBM
. (6.39)
Employing the Kaluza-Klein ansatz (6.37) and using the partial derivative
∂M˜ = ∂µ ∂µ ∂M
(6.40) it is now straightforward to compute the fluxes ˜FAˆ˜Bˆ˜Cˆ˜ and ˜FAˆ˜ defined in (5.11) and (5.14).
After some algebra, we find the following non-vanishing components F˜ˆaˆbˆc =eaˆdeˆb
eeˆcf 3 D[dBef]+AbD[dDeAbDf] F˜ˆaˆb ˆ
c= 2e[ˆadDdeˆb]
eeeˆc= ˜fˆaˆcˆb F˜ˆaˆbCˆ =−eˆadeˆbeEbCDˆ 2D[dAbDe] F˜ˆaBˆCˆ =eˆadDdEbBˆDEbCDˆ
F˜ˆa = ˜fˆaˆˆcc+ 2eaˆbDbφ . (6.41)
As we want to determine the full covariant fluxes Fˆ˜
ABˆ˜Cˆ˜, we also have to take the back-ground contribution FAˆ˜Bˆ˜Cˆ˜ into account. Since the background vielbein (6.39) only de-pends on internal coordinates, the only non-vanishing components of FA˜B˜C˜ are given by
FABC = 2Ω[AB]C. (6.42)
Thus, they yield the non-vanishing components Fˆaˆbˆc =−eˆadeˆb
eeˆcfAbdDAbeEAbfFFDEF FˆaˆbCˆ =eaˆdeˆb e
AbcDAbdEEbCˆ
FFDEF FˆaBˆCˆ =−eˆabAbbDEbBˆ
E
EbCˆ
FFDEF FAˆBˆCˆ =EAˆ DEBˆ
EECˆ
FFDEF . (6.43) Combining these results with (6.41) and keeping the gauge covariant quantities in mind
DbµEbAˆ
B =∂µEbAˆ
B−FBCDAbµCEbAˆ D
FbAµν = 2∂[µAbν]A−FABCAbµBAbνC
Gbµνρ = 3∂[µBνρ]+Ab[µA∂νAbρ]A−FABCAbµAAbνBAbρC, (6.44) which were discussed in section 6.2, we finally arrive at the desired result
Faˆˆbˆc=eˆaµeˆbνeˆcρGbµνρ Fˆaˆbˆc= 2e[ˆaµ∂µeˆb]νeνˆc FˆaˆbCˆ =−eaˆµeˆb
ν
EbCAˆ FbAµν FˆaBˆCˆ =eˆaµDbµEbBˆ A
EbCAˆ
FAˆBˆCˆ =EbAˆ D
EbBˆ E
EbCˆ
F FDEF Fˆa= ˜fabb + 2eˆaµ∂µφ . (6.45)
6.3. DFT on group manifolds All gauge covariant objects carry indices such as A, B, C, . . . instead ofI, J, K, . . . in this context, as opposed to the last section. The reason for this lies in the fact that they have to carry O(n, n) indices, which are the former in DFTWZW (depicted in (5.4)) and the latter in the original formulation. From this point on, all remaining computations proceed in the same manner as explained in the last section. Consequently, we obtain the effective action
Seff = Z
dD−nx√
−g e−2φ
R+ 4∂µφ ∂µφ− 1
12GbµνρGbµνρ
−1
4HbABFbAµνFbBµν+ 1
8DbµHbABDbµHbAB−V
, (6.46) as well, when plugging our results into the action (5.33) of DFTWZW’s flux formulation.
As previously explained, we just need to replace the indices I, J, K, . . . by A, B, C, . . ..
However, this is for conventional purposes only. On top of that, the scalar potential in DFTWZW
V =−1
4FACDFBCDHbAB+1
2FACEFBDFHbABHbCDHbEF (6.47) lacks the strong constraint violating term 1/6FABCFABC. It appears as a cosmological constant in gauged supergravities, even if the strong constraint is not imposed on the background field. We already argued in section 5.2.1 why this term does not occur in our formulation. Anyway, it is totally legitimate to add it by hand, as it was done in the original flux formulation, to the action from a bottom up perspective. It does not spoil any of the theory’s symmetries.
This new approach solves an ambiguity of generalized Scherk-Schwarz compactifi-cations: In the DFTWZW framework, the twist is equivalent to the background gen-eralized vielbein EAI. Its construction works in the same fashion as for conventional Scherk-Schwarz reductions. The reason for it lies in the appearance of standard 2D-diffeomorphisms which are absent in the original DFT formulation. As a result, all math-ematical tools known for group manifolds are applicable. However, all these features are immediately lost upon returning to traditional DFT by imposing the extended strong constraint. The extended strong constraint, necessary for this transition, breaks the 2D-diffeomorphism invariance. Thus, we are left with the issues outlined in section 6.2 [4].
All derivations performed in DFTWZW so far are top down. We began with the full bosonic CSFT [2, 3] and reduced it step by step until we finally reached the low energy effective action (6.46). Hence, it is immediately possible to check the uplift of solutions for the equation of motion to full string theory. We only have to keep in mind that all results obtained so far are only valid at tree level. Requiring further consistency at loop level, e.g. a modular invariant partition function, puts further additional restrictions upon the theory. We can learn even more from the CSFT perspective: The background fluxesFABC scale with 1/√
k, where k denotes the level of the Kaˇc-Moody algebra on the worldsheet.
6. Generalized Scherk-Schwarz Compactifications in DFT on Group Manifolds
Moreover, we can decompose them according to FABC = 1
√
kfABC (6.48)
and assume that the structure coefficients fABC are normalized, e.g.
fACDfBDC = 1
2δAB. (6.49)
As a consequence, the gauge covariant derivative becomes DbµVA=∂µVA− 1
√kfABCAbµBCC. (6.50) Following this equation, we directly read off the Yang-Mills coupling constant
gYM = 1
√
k . (6.51)
It should be noted that the geometric interpretation of DFTWZW only holds for the large level limit k 1. The corresponding effective theory is then weakly coupled and hence can be treated perturbatively. Nevertheless, freezing out all fluctuations in the internal directions, which happens for generalized Scherk-Schwarz compactifications, extends our results to k = 1. This case requires to reduce the number of external directions to cancel the total central charges of the bosons and the ghost system [4].