Einstein-Cartan Calculus for Exceptional Geometry

0.29 MB·

by  Hadi Godazgar

DAMTP-2014-9
AEI-2014-000
Einstein-Cartan Calculus for Exceptional Geometry
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n Hadi Godazgar⋆ , Mahdi Godazgar† and Hermann Nicolai‡
a
J
⋆ DAMTP, Centre for Mathematical Sciences,
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2 University of Cambridge,
Wilberforce Road, Cambridge,

]
h CB3 0WA, UK
t
-
p
Max-Planck-Institut fu¨r Gravitationsphysik,
e ‡
h Albert-Einstein-Institut,
[ Am Mu¨hlenberg 1, D-14476 Potsdam, Germany

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v ⋆[email protected], †[email protected],
4 [email protected]
‡
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January 29, 2014
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Abstract
i
X
In this paper we establish and clarify the link between the recently found E generalised geo-
r 7(7)
a metric structures, which are based on the SU(8) invariant reformulation of D =11 supergravity
proposed long ago, and newer results obtained in the framework of recent approaches to gener-
alised geometry, where E duality is built in and manifest from the outset. In making this
7(7)
connection, the so-called generalised vielbein postulate plays a key role. We explicitly show how
thispostulatecanbe usedtodefineanE valuedaffineconnectionandanassociatedcovariant
7(7)
derivative, which yields a generalised curvature tensor for the E based exceptional geometry.
7(7)
The analysis of the generalised vielbein postulate also provides a natural explanation for the
emergence of the embedding tensor from higher dimensions.
1 Introduction
Recent progress [1], along the lines of an older proposal [2], on understanding the extent to which
the E Cremmer-Julia duality symmetry [3, 4] is inherent to the full D = 11 supergravity theory
7(7)
[5] has lead to a new formulation of the D = 11 theory, which apart from pointing to new geometric
structures in eleven dimensions, provides an appropriate framework in which to address questions
regarding the relation between D = 11 supergravity and four-dimensional maximal gauged super-
gravity theories [6, 7]. In this paper, we will clarify the relation of these results to more recent
approaches to generalised geometry, especially [8, 9, 10], and show how a synthesis of the different
approaches emerges.
The formalism of Ref. [1] is based on the SU(8) invariant reformulation of D = 11 supergrav-
ity [2], in which the local and global gravitational symmetries of the eleven-dimensional theory are
abandoned and one performs a 4+7 split of all fields in the theory. Importantly, dependence on all
eleven coordinates is retained throughout and one remains on-shell equivalent to the original theory
throughouttheconstruction. Anessential characteristic oftheanalysisofRef.[2],andamaindistin-
guishing feature in comparison with more recent work, is the use of supersymmetry transformations
to find new SU(8) and E structures in the eleven-dimensional theory. The most significant such
7(7)
structures are the “generalised vielbeine” [2, 11, 1], which replace the eleven-dimensional fields that
would contribute to scalar degrees of freedom in a reduction to four dimensions. As in [2], these are
derived by considering the supersymmetry transformation of eleven-dimensional fields that would
contribute to vector degrees of freedom in a reduction to four dimensions. A crucial ingredient in
constructing the full set of “generalised vielbeine” is to consider dual fields in eleven dimensions.
These building blocks are to be viewed as the components of a single E 56-bein that we shall
7(7)
V
henceforthsimplyrefertoasthe“generalisedvielbein”,inanalogywiththeterminologyusedinmore
recent literature [12]. In particular, the generalised vielbein as derived directly from the D = 11
theory in [1] coincides with the generalised vielbein that lies at the heart of other recent approaches
to generalised geometry [13] (see also [12]), where it is constructed from the E /SU(8) coset using
7(7)
an algebraic method known as non-linear realisation [14, 15, 16]. More recently, the generalised
geometry ideas that have been used to describe the seven-dimensional sector of D = 11 supergravity
in a 4+7 split have been extended to incorporate the four-dimensional part, in this way arriving at
an E covariant extension of the whole theory [17, 10].
7(7)
An important aspect of the formalism developed in [1] is the fact that the components of the
generalised vielbein satisfy differential constraints [2, 1] – called ”generalised vielbein postulates”
(GVPs) due to their resemblance to the usual vielbein postulate in differential geometry. It should
be emphasised that here these equations are not postulated, but follow directly from the explicit
expressions for the generalised vielbein in terms of the various fields and dual fields of D = 11
supergravity. In this sense, the present approach is ‘bottom up’, in contrast to other approaches,
where similar relations follow from more abstract geometrical reasoning. One of our main results
here is to show how these ingredients can be used to develop an Einstein-Cartan calculus that is
largely analogous to the one for the standard vielbein.
The GVPs divide into two sets: those in which the derivative acting on the component of the
generalised vielbein is taken with respect to the D = 4 directions and those in which the derivative
is with respect to the D = 7 directions. Using a terminology where “external” refers to D = 4 in
the 4+7 split of D = 11, and “internal” refers to D = 7, even though we remain on-shell equivalent
to the D = 11 theory and no reduction is assumed, we refer to the former set as “external GVPs”
and the latter set as “internal GVPs”. The GVPs are important in establishing a link between the
D = 11 theory and D = 4maximal gauged theories derived as a reduction thereof. Inparticular, the
external GVPs can be regarded as providing a higher dimensional origin of the embedding tensor
1
[18, 19, 20, 21, 22], as has been explicitly demonstrated for the S7 reduction [6] and Scherk-Schwarz
flux compactifications [7]. The relationship between D = 11 supergravity and D = 4 supergravity
is an important aspect of the SU(8) invariant reformulation of the D = 11 theory [2], and recent
developments therefrom [11, 1], in, for example, establishing non-linear ansa¨tze [23, 11, 24, 6] and
consistency of the S7 reduction [25, 26]. Very recently, this aspect has also been studied in Ref. [27]
where the generalised vielbein is related by a generalised Scherk-Schwarz ansatz to the E matrix
d(d)
parametrised by the scalars of maximal gauged supergravity. This allows them to verify/conjecture
non-linear ansa¨tze for various sphere reductions. The validity of the new ansa¨tze can be established
by an analysis along the lines of Refs. [2, 23, 11, 6] for the appropriate sphere reductions.
In this paper, we return to the reformulation of D = 11 supergravity developed in [1] and
proceed to make concrete the indications that there is an E generalised geometry underlying
7(7)
the constructions there. In particular, we make contact with recent results in duality-manifest
based approaches to generalised geometry [8, 9] 1 that have focused on similar issues from a duality
group perspective. We condense all the objects and equations, in particular the GVPs, into an
E covariant form such that the previous expressions can be obtained as particular components
7(7)
of the new expressions under SL(8) and GL(7) decompositions of E . Thus, even though general
7(7)
covariance in D = 11 has been abandoned in the 4+7 split, we obtain a reformulation that has
general covariance in the D = 4 directions and a “generalised general covariance” based on E in
7(7)
the D = 7 space in a manner consistent with the results of Ref. [8, 9, 10].
A prerequisite for introducing E covariance, and thus replacing GL(7) indices with E
7(7) 7(7)
indices, is that the seven-dimensional space on which the generalised geometry is constructed ap-
parently requires an extension to a 56-dimensional space 2 such that the seven internal coordinates
ym are extended to a set of 56 internal coordinates y , where labels the 56 representation
M
{ } { } M
of E [13]. However, in order for the geometric structures, such as the algebra of generalised dif-
7(7)
feomorphisms, to be consistent one must impose a constraint, the section condition, that ultimately
reduces the enlarged space to an at most seven-dimensional space [8, 9]. While the necessity of such
a restriction is plainly evident from the fact that no consistent supergravity appears to exist beyond
eleven dimensions, its necessity can also be seen from a more geometrical perspective: supposing
that the generalised vielbein did depend on 56 internal coordinates, we would have the textbook
V
formula
∂y
′N
(y) = (y )
′ ′
VM VN ∂yM
for the transformation under arbitrary diffeomorphisms in 56 dimensions. However, the transition
matrix ∂y /∂y being an element of GL(56), this operation would throw the 56-bein out of the
′M N
V
cosetE /SU(8). Onemightthereforeaskwhetherthereexistsasetofrestricteddiffeomorphismsin
7(7)
56 dimensions, such that ∂y /∂y E and the transformed generalised vielbein remains in the
′M N 7(7)
∈
coset. However, this possibility is excluded by Cartan’s Theorem, according to which there do not
exist ‘exceptional algebras of vector fields’ on manifolds, the only possibilities being (essentially) the
algebras of ordinary diffeomorphisms, volume preserving diffeomorphisms and symplectomorphisms
[36, 37] (see also Ref. [38]). Similar comments apply to the 3+8 split associated to the E duality
8(8)
group, as already noted in [39].
In section 2, wereview the requiredresultsfrom [1], rewritingthem in amannerthatmakes their
E structure manifest. We rewrite the GVPs, in section 3, using the E structures defined in
7(7) 7(7)
section 2. Then, in section 4, we explicitly demonstrate how the coordinate and gauge transforma-
tions of the generalised vielbein can be packaged into a single transformation given by generalised
1Forfurther references see [28, 29, 30, 31, 32, 33, 34, 35].
2Moreprecisely,the11-dimensionalspace-timemanifoldwouldhavetobeextendedtoa(4+56)-dimensionalspace,
but we can ignore thedependenceon the four external coordinates for theargument to bepresented.
2
diffeomorphisms [8]. Finally, in section 5, we similarly package the GVPs into single E covariant
7(7)
equations. The equation corresponding to the external GVPs is precisely of the same form as the
Cartan equation in four-dimensional maximal gauged theories, allowing us to identify the higher di-
mensional object, an operator, that gives the embedding tensor upon reduction to four dimensions.
On the other hand, the internal GVP is the generalised geometric analogue of the vielbein postu-
late and yields the generalised connection for the generalised geometry. We give the transformation
properties of the generalised connection. Furthermore, we find that a covariant derivative defined
using the generalised connection transforms as a generalised tensor density of weight 1/2 less than
the weight of the generalised tensor on which the covariant derivative acts. Thus, a generalised Rie-
mann curvature tensor obtained by commuting two covariant derivatives transforms as a generalised
tensor density ofweight 1.Weexplicitly presentthecomponentsof thegeneralised Riemanntensor
−
and note that it is indeed generalised gauge covariant.
The conventions used in this paper are the same as those of Ref. [2]. In particular, M,N,...
and A,B,... denote eleven-dimensional spacetime and tangent space indices, respectively. Indices
A,B,... are also used as SU(8) indices. However, it should be clear from the context what type of
index is being referred to. Similarly, µ,ν,... and α,β,..., and m,n,... and a,b,... denote D = 4
and D = 7 spacetime and tangent space indices, respectively.
2 Preliminaries
2.1 Generalised vielbein
As explained in much detail in our previous work [1], a generalised vielbein , which can be viewed
V
as a 56-bein of E , can be defined directly in eleven dimensions. This 56-bein depends on the
7(7)
fields and on the dual fields of D = 11 supergravity, as obtained by performing a 4+7 split on the
original fields, with all fields still depending on all eleven-dimensional coordinates. In particular, it
depends on the siebenbein e a, which is obtained from a 4+7 decomposition of the original elfbein
m
of D = 11 supergravity in a triangular gauge (which breaks the original tangent space symmetry
SO(1,10) of the theory to SO(1,3) SO(7)):
×
∆ 1/2e α B me a
E A(x,y) = − ′µ µ m , ∆ dete a. (1)
M m
0 ema ! ≡
Here, as usual, we splitthe eleven-dimensional coordinates zM into fourexternal coordinates xµ
{ } { }
and seven internal coordinates ym . The 3-form and 6-form gauge fields, on which the 56-bein also
{ }
depends, are linked via the duality relation
√2
F = 7!D A +7! A D A
M1···M7 [M1 M2···M7] 2 [M1M2M3 M4 M5M6M7]
√2
iǫ Ψ Γ˜M8 M11RSΨ +12ΨM8Γ˜M9M10ΨM11 (2)
− 192 M1···M11 R ··· S
(cid:16) (cid:17)
in eleven dimensions, from which all pertinent relations linking the 4-form and 7-form field strengths
can be obtained by choosing the indices appropriately. Although we will ignore the fermionic terms
in this duality relation in the remainder, it should be clear that this duality relation introduces a
hidden dependence of the 56-bein (which we are about to present) on the fermionic fields as well.
A main result of [1] is thus the complete identification of the 56-bein in terms of the siebenbein,
and the internal components of the 3-form and the 6-form, such that (e,A(3),A(6)). With
V ≡ V
3
proper E normalisation, the components of the generalised vielbein are explicitly given by
7(7)
√2
m = ∆ 1/2Γm , (3)
V AB − 8 − AB
√2
= ∆ 1/2 Γ +6√2A Γp , (4)
VmnAB − 8 − mnAB mnp AB
(cid:16) (cid:17)
√2 1
mn = ηmnp1 p5∆ 1/2 Γ +60√2A Γ
V AB − 8 · 5! ··· − " p1···p5AB p1p2p3 p4p5AB
√2
6!√2 A A A Γq , (5)
− qp1···p5 − 4 qp1p2 p3p4p5 AB#
(cid:16) (cid:17)
√2 1
= ηp1 p7∆ 1/2 (Γ Γ ) +126√2 A Γ
VmAB − 8 · 7! ··· − " p1···p7 m AB mp1p2 p3···p7AB
√2
+3√2 7! A + A A Γ
× mp1···p5 4 mp1p2 p3p4p5 p6p7AB
(cid:16) (cid:17)
9! √2
+ A + A A A Γq , (6)
2 mp1···p5 12 mp1p2 p3p4p5 p6p7q AB#
(cid:16) (cid:17)
where Γm em Γa are the D = 7 gamma matrices with seven-dimensional curved indices and
a
≡
ηm1...m7 is the seven-dimensional permutation symbol (tensor density of weight +1). These ex-
pressions are obtained by insisting on the E covariance of the supersymmetry variations (after
7(7)
appropriate field redefinitions), see also remarks in section 2.2 below.
The vielbein is subject to local SU(8) rotations (depending on all eleven coordinates), such that
the above expressions in terms of quantities of D = 11 supergravity correspond to a special gauge
choice, as explained already in [2]. Furthermore, complex conjugation raises (lowers) SU(8) indices
MNAB ( MNAB)∗ , MNAB ( MNAB)∗, (7)
V ≡ V V ≡ V
where we have combined the GL(7) indices m,n,... into SL(8) indices M,N,... according to 3
MN mn, m8 , MN mn, m8 . (8)
V ≡ V V V ≡ V V
(cid:0) (cid:1) (cid:0) (cid:1)
That is, complex conjugation only affects the SU(8) indices. We will also use proper E indices
7(7)
, ,... corresponding to the 56 representation, such that
M N
MN MN
MN, , M = ΩMN , MN , (9)
VM ≡ V V V VN ≡ V −V
(cid:0) (cid:1) (cid:0) (cid:1)
where the components of the symplectic form Ω are
MN
MN MN PQ PQ
Ω PQ = δPQ, ΩMN = δMN,
−
MNPQ
ΩMNPQ = 0, Ω = 0, (10)
3Forbrevity,we will often use thesimplifying notation Vm ≡Vm8 =−V8m and Bm ≡Bm8 =−B8m,etc.
4
and Ω is given by Ω Ω = δ . Moreover, with the above normalisation, satisfies the
MP M
E pMroNperties NP N V
7(7)
AB AB = iΩ ,
AB AB
VM VN −VM VN MN
Ω AB = iδAB,
MNVM VNCD CD
Ω AB CD = 0, (11)
MN
VM VN
which can be directly verified from definitions (3)–(6). The generalised vielbein also satisfies the
following E covariant supersymmetry transformation
7(7)
δ = √2Σ CD, (12)
AB ABCD
VM VM
with the complex self-dual SU(8) tensor
Σ = ε¯ χ + 1ǫ ε¯EχFGH. (13)
ABCD [A BCD] 4! ABCDEFGH
In the SU(8) invariant reformulation, the D = 11 gravitino Ψ is rewritten in terms of SU(8)
M
covariant chiral fermions ϕ A and χ and their complex conjugates ϕ ,χABC [4, 2]. Theprecise
µ ABC µA
relation between (12) and the D = 11 supersymmetry variations also involves an SU(8) rotation
that we have dropped.
In addition to local SU(8) transformations, the generalised vielbein is subject to several gauge
transformations which it inherits from the fields on which it depends, to wit, internal diffeomor-
phisms, and the tensor gauge transformations associated to the 3-form and the 6-form gauge poten-
tials. Recall that in our scheme all transformation parameters depend on eleven coordinates. The
transformations under internal diffeomorphisms are straightforward to obtain:
1
δ m = ξp∂ m ∂ ξm p ∂ ξp m ,
AB p AB p AB p AB
V V − V − 2 V
1
δ = ξp∂ 2∂ ξp ∂ ξp ,
mnAB p mnAB [m n]pAB p mnAB
V V − V − 2 V
1
δ mn = ξp∂ mn +2∂ ξ[m n]p + ∂ ξp mn ,
AB p AB p AB p AB
V V V 2 V
1
δ = ξp∂ + ∂ ξp + ∂ ξp . (14)
mAB p mAB m pAB p mAB
V V V 2 V
Note that the density terms come from the overall factor of ∆ 1/2 in the definition of . With
±
VM
respect to the tensor gauge transformations, we have
δA = 3!∂ ξ , δA = 3√2∂ ξ A (15)
mnp [m np] mnpqrs [m np qrs]
and
δA = 0 , δA = 6!∂ ξ (16)
mnp mnpqrs [m npqrs]
with the 2-form and 5-form gauge parameters ξ and ξ , respectively. Substituting these
mn mnpqr
transformations into the explicit expressions for the generalised vielbein components in (3)–(6), it is
straightforward to deduce the transformation properties
δ m = 0, δ = 36√2∂ ξ p ,
AB mnAB [m np] AB
V V V
δ mn = 3√2ηmnpqrst∂ ξ , δ = 18√2∂ ξ np , (17)
AB p qr stAB mAB [m np] AB
V V V V
5
and
δ m =δ = 0, δ mn = 6 6!√2ηmnp1 p5∂ ξ q ,
V AB VmnAB V AB − · ··· [q p1···p5]V AB
δ = 3 6!√2ηn1 n7∂ ξ . (18)
VmAB · ··· [m n1···n5]Vn6n7AB
We already see here that these transformation parameters can be nicely combined as
Λ Λm,Λ ,Λmn,0 (19)
M mn
≡
whereΛm ξm, Λ ξ andΛmn ηmn(cid:0)p1 p5ξ (thep(cid:1)recisecoefficientswillbeconveniently
∼ mn ∼ mn ∼ ··· p1···p5
chosen later). In this way ordinary diffeomorphisms and tensor gauge transformations are unified
into asingle setof transformations. This will beshown explicitly in section 4, wherewewill consider
generalised diffeomorphisms and show how the above transformations can be compactly written in
terms of a single generalised Lie derivative, see equation (57). The ‘missing’ seven components Λ
m
in this identification are obviously associated with ‘dual’ internal diffeomorphisms, but will actually
be seen to drop out.
2.2 Vector fields
The components of the generalised vielbein can be obtained by considering the supersymmetry of a
set of eleven-dimensional fields with one D = 4 index [2, 11, 1]. As such they are known as vectors
in accord with the convention of using four-dimensional language for analogous D = 11 structures
adopted here. We similarly combine the vectors into a 56 of E
7(7)
MN
BµM = (Bµ , BµMN). (20)
The proper definitions of these 56 vector fields follow from the identifications
1
m = B m, = 3√2 A B pA ,
µ µ µmn µmn µ pmn
B −2 B − −
(cid:0) (cid:1)
√2
mn = 3√2ηmnp1...p5 A B qA A B qA A
Bµ − µp1···p5 − µ qp1···p5 − 4 µp1p2 − µ qp1p2 p3p4p5!
(cid:0) (cid:1)
= 18ηn1...n7 A +(3c˜ 1)(A B pA )A
Bµm − µn1...n7,m − µn1...n5 − µ pn1...n5 n6n7m

√2
+c˜A (A B pA )+ (A B pA )A A , (21)
n1...n6 µn7m− µ pn7m 12 µn1n2 − µ pn1n2 n3n4n5 n6n7m
!
where c˜is an undetermined constant. These are related to the generalised vielbein via the following
supersymmetry transformation [2, 11, 1]
δ = iΩ 2√2εAϕB +ε γ χABC + h.c. (22)
BµM MNVNAB µ C µ
(cid:16) (cid:17)
6
using the supersymmetry transformations of the fields given in [2, 11, 1]. In particular [1]
1 √2c˜
δA = εΓ˜ Ψ 8εΓ˜ Γ˜ Ψ + εΓ˜ Ψ A
µm1...m7,n −9! µm1...m7 n− n [µm1...m6 m7] 5! [µm1...m4 m5 m6m7]n
(cid:16) (cid:17)
√2 √2
+ εΓ˜ Ψ A + A A
3 [µm1 m2 m3...m7]n 12 m3...m5 m6m7]n
!
√2
√2c˜εΓ˜ Ψ A + A A , (23)
− [µm1 m2 m3...m7]n 4 m3...m5 m6m7]n
!
where Ψ is the component of the D = 11 gravitino along the internal directions (prior to any
m
redefinition).
The transformation of the components of under internal diffeomorphisms is
BµM
δ m = ξp∂ m ∂ ξm p,
µ p µ p µ
B B − B
δ = ξp∂ 2∂ ξp ,
µmn p µmn [m µn]p
B B − B
δ mn = ξp∂ mn + 2∂ ξ[m n]p+∂ ξp mn. (24)
µ p µ p µ p µ
B B B B
We note that mn transforms as a tensor density of weight 1 because of the tensor density η in its
µ
B
definition, (21). The transformation of under internal 2-form and 5-form gauge transformations
BµM
is
δ m = 0, δ = 36√2∂ ξ p
µ µmn [m np] µ
B B − B
δ mn = 3√2∆ǫmnp1 p5∂ ξ . (25)
Bµ − ··· p1 p2p3Bµp4p5
and
δ m = δ = 0, δ mn = 6 6!√2ηmnp1 p5∂ ξ q. (26)
Bµ Bµmn Bµ − · ··· [q p1···p5]Bµ
Since we do not know at this point how A transforms under coordinate, 2-form and 5-
µm1...m7,n
form gauge transformation we cannot, yet, determine the gauge transformation rule for the final
component . Let us nevertheless anticipate the results of section 4, where we will find the
µm
B
transformation rule from the E structure of internal coordinate and gauge transformations:
7(7)
δ = ξp∂ + ∂ ξp +∂ ξp mn, (27)
µm p µm m µp p µ
B B B B
δ = 18√2∂ ξ pq, δ = 3 6!√2ηn1 n7∂ ξ , (28)
Bµm − [m pq]Bµ Bµm · ··· [m n1···n5]Bµn6n7
for coordinate, 2-form and 5-form gauge transformations, respectively. Going backwards from these
expressions, we can deduce that A transforms as a tensor under internal coordinate trans-
µn1...n7,m
formations and under 2-form and 5-form gauge transformations it transforms as:
δA = 18c˜∂ ξ A +√2(9c˜ 2)∂ ξ A A
µn1...n7,m − [m n1n2] µn3...n7 − n1 n2n3 µn4n5 mn6n7
(9c˜ 2)
− ∂ ξ A A , (29)
− √2 [m n1n2] µn3n4 n5...n7
δA = 6!(3c˜ 1)∂ ξ A , (30)
µn1...n7,m − − [m n1...n5] µn6n7
respectively. Here c˜is the undetermined constant that appeared already in [1], and that is also not
fixed by imposing E covariance. As for the generalised vielbein, we will show that the formulae
7(7)
(25),(26)and(28),togetherwiththeactionofinternaldiffeomorphisms,canbecompactlyassembled
into a single E covariant formula, (67).
7(7)
7
3 Generalised vielbein postulate
The generalised vielbeine satisfy differential constraints along the four external and the seven inter-
nal directions, which are called generalised vielbeine postulates (GVPs) in analogy with the usual
vielbein postulate in differential geometry. These constraints are identities that can be directly ver-
ified from the explicit expressions given above, just like the usual vielbein postulate is an identity
when the affine connection and the spin connection are expressed in terms of the usual vielbein.
The external GVPs, which are the GVPs along the d = 4 directions are of the form 4
∂ m + C m +2 nD m 2D m n D n m = mCD, (31)
µV AB Qµ[AV B]C Bµ nV AB − nBµ V AB − nBµ V AB PµABCDV
∂ + C +2 pD 4D p D p
µVmnAB Qµ[AV|mn|B]C Bµ pVmnAB − [mB|µ| Vn]pAB − pBµ VmnAB
+6D p = CD, (32)
[m µ np] AB µABCD mn
B| | V P V
∂ mn + C mn +2 pD mn +6D [m np] D p mn
µV AB Qµ[AV B]C Bµ pV AB pBµ V AB − pBµ V AB
1
+ ηmnp1...p5D +4D p[m n] = mnCD, (33)
2 p1Bµp2p3Vp4p5AB pBµ V AB PµABCDV
∂ + C +2 pD +2D p +D p
µVmAB Qµ[AVmB]C Bµ pVmAB mBµ VpAB pBµ VmAB
+3D pq 2D pq = CD, (34)
[m µpq] AB p µ qmAB µABCD m
B| | V − B V P V
where D is the covariant derivative with respect to seven-dimensional diffeomorphisms, e.g.
m
D n ∂ n+Γn p (35)
mBµ ≡ mBµ mpBµ
p
with the internal affine connection Γ . In GVPs, above, the combination of components of the
mn
vector field and the generalised vielbein in each term is exactly such that the discrepancy in
µM
B
the weights of the components of the generalised vielbein is compensated by the differing weights in
the components of the vector field . Hence the weights of the terms in each GVP are consistent.
µM
B
Note that in previous work [2, 1] these relations were given without the affine connection terms,
but the relations above are still equivalent to the original ones (see [40]), as all terms containing the
affine connections cancel in the above relations, as well as the ones given below. The connection
coefficients are of the form
A = 1 em D B ne (ep e ) Γab √2e α F Γabc η FβγδaΓ , (36)
QµB −2 a m µ nb− aDµ pb AB − 12 µ αabc AB − αβγδ aAB
h i
(cid:0) (cid:1)
= 3 em D B ne (ep e ) Γa Γb √2e αF Γa Γbc
PµABCD 4 a m µ nb− aDµ pb [AB CD]− 8 µ abcα [AB CD]
h i
√2e ηαβγδF Γ Γab , (37)
− 48 µα aβγδ b[AB CD]
where
∂ B mD ∂ +2 mD . (38)
µ µ µ m µ µ m
D ≡ − ≡ B
4NotethatthesigninfrontoftheP structuresinboththeexternalandinternalGVPsisoppositetowhatappears
in the GVPs as written in Ref. [1]. This is because of a differing definition of the generalised vielbein V—more
specifically, an extrafactor of iin thedefinition of V.
8
In the dimensionally reduced theory, the kinetic term for the scalar fields is ABCD µ ,
∝ Pµ PABCD
while the ‘composite’ SU(8) connection A is required for the covariantisation of the fermionic
µ B
Q
couplings.
Similarly, the generalised vielbein satisfies a GVP along the internal directions. The relevant
relations were derived in [1] and read
1
∂ m +Γm n + Γn m + C m = mCD, (39)
pV AB pnV AB 2 pnV AB Qp[AV B]C PpABCDV
1
∂ +2Γq + Γq 6√2Ξ q + C
pVmnAB p[mVn]qAB 2 pqVmnAB − p|mnqV AB Qp[AVmnB]C
= CD, (40)
pABCD mn
P V
1
∂ mn 2Γ[m n]q Γq mn 6√2ηmnq1 q5Ξ q6
pV AB − pqV AB − 2 pqV AB − ··· p|q1...q6V AB
1
ηmnq1 q5Ξ + C mn = mnCD, (41)
− √2 ··· p|q1q2q3Vq4q5AB Qp[AV B]C PpABCDV
1
∂ Γn Γq √2ηn1 n7Ξ 3√2Ξ rs
pVmAB − pmVnAB − 2 pqVmAB − ··· p|n1···n6Vn7mAB − p|rsmV AB
+ C = CD, (42)
Qp[AVmB]C PpABCDVm
where the first few terms in each of the above equations correspond to the general covariant deriva-
tive, i.e.
1
D n ∂ n + Γn p + Γp en , (43)
mVAB ≡ mVAB mpVAB 2 mp AB
1
D ∂ + 2Γq + Γp en , (44)
pVmnAB ≡ pVmnAB p[mVn]qAB 2 mp AB
andsoon. Notethatthecomponentsofthegeneralised vielbeinaredensitieswithrespecttointernal
coordinate transformations, hence the extra terms involving Γn . Furthermore, the connection
mn
coefficients A and are
QmB PmABCD
A = 1ω Γab + √2ife Γa √2e aF Γbcd, (45)
QmB −2 mab AB 14 ma AB − 48 m abcd AB
= √2ife aΓ Γb + √2e aF Γb Γcd , (46)
PmABCD 56 m ab[AB CD] 32 m abcd [AB CD]
where
f = 1 iηαβγδF = 1ηa1...a7F . (47)
−24 αβγδ −7! a1...a7
The above connection coefficients can also be written in a more suggestive form
A = 1ω Γab + √2 F Γa1...a6 √2F Γabc, (48)
QmB −2 mab AB 146! ma1...a6 AB − 48 mabc AB
·
= √2 F Γa1 Γa2...a6 + √2F Γa Γbc , (49)
PmABCD −565! ma1...a6 [AB CD] 32 mabc [AB CD]
·
whence it is clear that they are invariant under 2-form and 5-form gauge transformations. The
expressions for A and given here differ from the expressions given before 5 because of
QmB PmABCD
5Seeequations (3.33) and (3.34) of Ref. [2].
9