7. Generalized Parallelizable Spaces from Exceptional Field Theory 109
7.1.3. Generalized Lie derivative
In a similar fashion to the SC (7.20), one can write the generalized Lie derivative for different EFTs, given in tab. 7.1, using the canonical form
LξVM =LξVM +YM NLK∂NξLVK, (7.25) with the Y-tensor and the standard Lie derivative on the extended space [7]. Once the SC is imposed, the infinitesimal generalized diffeomorphisms governing the gauge algebra close [123] according to
[Lξ
1,Lξ
2]VM =L[ξ
1,ξ2]EVM with [ξ1, ξ2]E= 1 2 Lξ
1ξ2− Lξ
1ξ2
. (7.26) It should be mentioned that this formulation of the gauge algebra also includes the DFT results for a specific choice of the Y-tensor, i.e. YM NLK = ηM NηLK, and therefore nat-urally extends to the EFT framework with the Y-tensor in SL(5) taking on the form in (7.21). Thus, it is serving as the logical starting point for our discussion. Furthermore,
7.1. Generalized Diffeomorphisms on Group Manifolds one should keep the rough structure necessary for the closure in mind, as we have to repeat these steps with a covariant derivative instead of a partial one later on. Computing
[Lξ1,Lξ2]VM − L[ξ1,ξ2]EVM, (7.27) we are left with sixteen different terms. All containing two partial derivatives. Although, the partial derivatives acts on the same variable only for four terms. On top of that, the Y-tensor exhibits the following properties
δF(BYAC)DE−Y(ACF GYB)GDE = 0 and
δ(FBYACDE)−YACG(FYBGDE)= 0 (7.28) which implies that for d ≤ 6 only terms annihilated by the SC remain. For U-duality groups with d > 6 the closure calculation becomes much more complicated [123]. Here, we are merely interested in a proof of concept for them. As a result, let us concentrate on the simplest cases and adjourn the rest to future work. The generalized and standard Lie derivative coincide for arbitrary scalars which yields
Lξs=Lξs . (7.29)
Applying the Leibniz rule we derive the action of generalized diffeomorphisms on one-forms
LξVM =LξVM −YP QN M∂QξNVP , (7.30) which are the dual objects of the vector representation. Subsequently, we have to remem-ber that YM NP Q needs to remain invariant under the generalized Lie derivative, i.e.
LξYM NP Q= 0. (7.31)
It is totally analogous to the statement in DFT where ηIJ stays invariant and therefore the O(d, d) is preserved. Hence, we have completed the list of requirements necessary to make EFT’s generalized diffeomorphisms compatible with standard diffeomorphisms.
Let us take a first step towards this direction. We change to flat indices while replacing all partial derivatives appearing in (7.25) by covariant ones and obtain
LξVA =ξB∇BVA−VB∇BξA+YABCD∇BξCVD. (7.32) This equation can be recast using flat derivatives
∇AVB=DAVB+ ΓACBVC and ∇AVB =DAVB−ΓABCVC (7.33) by introducing the spin connection ΓABC. Inserting this into the generalized Lie derivative gives us
LξVA=ξBDBVA−VBDBξA+YABCDDBξCVD +XBCAξBVC and
7. Generalized Parallelizable Spaces from Exceptional Field Theory
LξVA=ξBDBVA+VBDAξB−YCDBADDξBVC −XBACξBVC (7.34) with
XABC = 2Γ[AB]C+YCDBEΓDAE (7.35) where we collected all terms depending on the spin connection. Later, it will turn out that XABC is closely related to the embedding tensor known from gauged supergravities.
The Y-tensor should still remain invariant under the modified generalized Lie derivative.
It directly translates into the first linear constraint
∇CYABDE :=C1ABCDE = 2YF(ADEΓCFB)−2YAB(D|FΓC|E)F = 0 (C1) on the spin connection Γ after imposing DAYBCDE = 0. This constraint is a straight forward generalization of the metric compatibility (7.13) in DFTWZW.
In the next step, we demand closure of this adjusted generalized Lie derivative. All terms (7.27) spoiling the closure have to vanish analogously. We start with the ones incorporating no flat derivatives. These only disappear if the quadratic constraint
XBEAXCDE −XBDEXCEA+X[CB]EXEDA= 0 (7.36) holds. Analyzing these additional conditions makes it necessary to decompose XABC into a symmetric part ZCAB and an antisymmetric one by
XABC =ZCAB+X[AB]C. (7.37)
Furthermore, we see that all terms containing only one flat derivative acting on VA in (7.27) vanish, if we identify the torsion of the flat derivative with
[DA, DB] =X[AB]CDC. (7.38) Here, we have used DAXBCD = 0 andYABBC =Y(AB)(BC) which is only valid for d ≤6 in all computations. A consistent theory moreover requires the existence of a Bianchi identity. It takes on the form
[DA,[DB, DC]] + [DC,[DA, DB]] + [DB,[DC, DA]] = 0. (7.39) From explicitly evaluating the commutators above, we observe that this constraint is equivalent to the Jacobi identity
X[AB]EX[CE]D+X[CA]EX[BE]D +X[BC]EX[AE]D
DD = 0. (7.40) Antisymmetrizing (7.36) with respect to B, C, D yields
X[BC]EX[CE]A+X[DB]EX[CE]D+X[CD]EX[BE]D =−ZAE[BXCD]E (7.41)
7.1. Generalized Diffeomorphisms on Group Manifolds
and therefore is not zero. This leaves us with
ZAE[BXCD]EDA= 0 (7.42)
which in general does not vanish. For DFTWZW this is not the case sinceZABC vanishes and the issue does not occur. Thus, it is special to gEFT. As we show in subsection 7.1.4, the problem can be circumvented by reducing the dimension of the group manifold rep-resenting the extended space.
One of the important properties of the generalized Lie derivatives lies in the fact that the Jacobiator of its E-bracket only vanishes up to trivial gauge transformations. Hence, we want to take a closer look at them
ξA=YABCDDBχCD (7.43)
in the background of our modified generalized Lie derivative. Ultimately, we will benefit from it by being better able to organize terms appearing in the closure computation with one flat derivative operating either on ξ1 or ξ2. Plugging (7.43) into the generalized Lie derivative (7.34) yields the following relation
LξVA=C2aABCDEDBχCDVE +· · ·= 0 (7.44) where . . . refers to terms which vanish under the SC and as a result of the properties of the Y-tensor (7.28). The tensor
C2a CDEAB =YBFCDXF EA+ 1
2YAFCDX[F E]B+1
2YAFEHYGHCDX[F G]B (C2a) needs to vanish when trivial gauge transformations have the form given in (7.43).
Terms appearing with two derivatives in (7.27) become zero under the SC or due to (7.28). At this point, all terms we are left with contain one flat derivative acting on the gauge parameters ξ1 or ξ2. Since (7.27) is antisymmetric with respect to the gauge parameters, it is sufficient to check whether all contributions on one of the terms, e.g.
DAξ1B, vanish. They can be written in terms of the tensor
C2b CDEAB =ZADCδEB−ZBDEδCA−YBFECZADF +YABCFZFDE
+YABEFX[DC]F +YABCFX[DE]F −2YF(AECX[DF]B) = 0 (C2b) by
− 1
2C2a CDEAB +C2b CDEAB
DBξC1ξ2DVE = 0, (7.45) which obtains a contribution from trivial gauge transformations (C2a) as well. This is perfectly sensible as the E-bracket also only closes up to trivial gauge transformations.
7. Generalized Parallelizable Spaces from Exceptional Field Theory
However, generally there exists no explanation why the two contributions have to disap-pear independently. It implies that only the second linear constraint
−1
2C2a CDEAB +C2b CDEAB = 0 (C2)
has to be satisfied in combination with the first linear constraint (C1), and the quadratic constraint (7.36) for closure of generalized diffeomorphisms under the SC. Thus, one has to restrict the connection ΓABC in such a way that all these three restraints are fulfilled.
This outlines our steps during the next two subsections.
Right now, we can already perform a first consistency check of our results. Therefore, let us consider the O(d−1,d−1) T-duality group with
YABCD =ηABηCD, ΓABC = 1
3FABC and XABC =FABC. (7.46) In this case, the two linear constraints and quadratic constraint reduce to
C1ABCDE = 2
3ηDEFC(AB)− 2
3ηABFC(DE) = 0 (7.47)
C2a CDEAB =ηCD(FBEA+FAEB) = 0 (7.48)
C2b CDEAB =ηAB(FDCE+FDEC)−2ηECFD(AB) = 0 (7.49) as a consequence of the total antisymmetry of the structure constants FABC. Thus, this short computation is in perfect agreement with the closure of the gauge algebra known from DFTWZW [2,7].