Spheres, generalised parallelisability and consistent truncations
aa r X i v : . [ h e p - t h ] J a n Imperial/TP/14/DW/0ZMP-HH/14-3
Spheres, generalised parallelisability andconsistent truncations
Kanghoon Lee, a Charles Strickland-Constable b and Daniel Waldram ca,c Department of Physics, Imperial College London,Prince Consort Road, London, SW7 2AZ, UK a Center for Quantum Spacetime, Sogang University,Seoul 121-742, Korea b II. Institut f¨ur Theoretische Physik der Universit¨at Hamburg,Luruper Chaussee 149, D-22761 Hamburg, Germany
E-mail: [email protected] , [email protected] , [email protected] Abstract:
We show that generalised geometry gives a unified description of max-imally supersymmetric consistent truncations of ten- and eleven-dimensional super-gravity. In all cases the reduction manifold admits a “generalised parallelisation”with a frame algebra with constant coefficients. The consistent truncation then arisesas a generalised version of a conventional Scherk–Schwarz reduction with the framealgebra encoding the embedding tensor of the reduced theory. The key new result isthat all round-sphere S d geometries admit such generalised parallelisations with an SO ( d + 1) frame algebra. Thus we show that the remarkable consistent truncationson S , S , S and S are in fact simply generalised Scherk–Schwarz reductions. Thisdescription leads directly to the standard non-linear scalar-field ansatze and as anapplication we give the full scalar-field ansatz for the type IIB truncation on S . ontents GL + ( d + 1 , R ) generalised geometry 52.3 Spheres as generalised parallelisable spaces 92.4 Generalised SL ( d + 1 , R ) Scherk–Schwarz reduction on S d S and SO (3 ,
3) generalised geometry 123.1.1 Relation to gauged supergravity 143.1.2 Other parallelisations 153.2 S and E generalised geometry 163.3 S and E generalised geometry 163.3.1 Consistent truncations and the general scalar ansatz on S S and E generalised geometry 20 S d
28B Type IIB E generalised geometry 30C Generalised connections and conventional Scherk–Schwarz 33 Consistent truncations of gravitational theories are few and far between [1, 2]. Theclassic example is compactification on a local group manifold M = G/ Γ, where Γ isa discrete, freely-acting subgroup of a Lie group G . If the discrete group acts on theleft, the left-invariant vector fields ˆ e a define a global frame so M is parallelisable.Furthermore taking the Lie bracket[ˆ e a , ˆ e b ] = f abc ˆ e c (1.1)the coefficients f abc are constant. If in addition the “unimodular” condition f abb = 0is satisfied then one has a consistent truncation [3]. If the theory is pure metric, the– 1 –calar fields in the truncated theory come from deformations of the internal metric.One defines a new global frame ˆ e ′ a ( x ) = U ab ( x )ˆ e a (1.2)where U ab ( x ) depends on the uncompactified coordinates x . This frame defines thevielbein for the transformed metric. By construction the scalar fields U ab ( x ) parame-terised a GL ( d, R ) /O ( d ) coset. The truncated theory is gauged by the group G withthe Lie algebra given by the Lie bracket (1.1).More generally, as first considered by Scherk and Schwarz [3], any field theorycan be reduced on M using left-invariant objects, and by definition the resultingtruncation will be consistent. In particular, one can consider reductions of heterotic,type II or eleven-dimensional supergravity [4, 2, 5–7]. Since the parallelisation meansthe tangent space is trivial, M also admits global spinors and the truncated theorieshave the same number of supersymmetries as the original supergravity theory. Thestructure of such gauged supergravity theories is very elegantly captured by theembedding tensor formalism [8].In addition to these local group manifold reductions, there is a famous set ofremarkable consistent reductions on spheres, notably S [9] and S [10] for eleven-dimensional supergravity, S for type IIB (for which a subsector is known to beconsistent [11]), and S for the NSNS sector of type II supergravity [12]. However,generically reductions on coset spaces are not consistent and there is “no knownalgorithmic prescription” [2] for understanding the appearance of these few specialcases.In this paper we argue for a systematic understanding of consistent truncationsin terms of generalised geometry. In generalised geometry one considers structures onan generalised tangent space E . In the original formulation [13, 14] E ≃ T M ⊕ T ∗ M ,and the structure on E , together with the natural analogue of the Levi–Civita con-nection, capture the NSNS degrees of freedom of type II theories and the bosonic andfermionic equations of motion [15] (see [16] and also [17] for earlier geometric refor-mulations using the closely related Double Field Theory formalism [18]). There arealso other versions of generalised geometry [19–22] with structures and connectionswhich capture, for example, the full set of bosonic fields and equations of motionof type II and eleven-dimensional supergravity [23, 24]. The central point for us isthat in each case there is a direct generalised geometric analogue of a local groupmanifold, namely a manifold equipped with a global frame { ˆ E A } on E such that L ˆ E A ˆ E B = X ABC ˆ E C , (1.3)where X ABC are constant. By definition E is then trivial and we say the frame definesa “generalised parallelisation” of M [25]. Since E is trivial, the related generalisedspinor bundle [19] is also trivial and hence one also has globally defined spinors. Thus– 2 –e expect any truncated theory to have the same number of supersymmetries as theoriginal supergravity. Just as for the pure metric case, one can define a “generalisedScherk–Schwarz” reduction by defining a rotated generalised frameˆ E ′ A ( x ) = U AB ( x ) ˆ E B . (1.4)One is led to conjecture: Given a generalised parallelisation { ˆ E A } satisfying (1.3) there is a con-sistent truncation on M preserving the same number of supersymmetriesas the original theory with embedding tensor given by X ABC and scalarfields encoded by (1.4) . For compactifications on local group manifolds the conventional global frame { ˆ e a } always defines a generalised global frame, and this conjecture has already, at leastimplicitly, appeared in the literature [4, 7, 25]. In addition, without assuming a con-sistent truncation, the relation between the frame algebra and the embedding tensorof the reduced theory has been identified [26, 23, 27, 28] both in conventional gen-eralised geometry and in the language of Double Field Theory [18] and its M-theoryextensions [29]. The generalised Scherk–Schwarz ansatz (1.4) is also in practise used,for the metric components, in the original work on S [30, 9], and, recently, thishas been extended to all the flux components [31]. In [32] the four-dimensional em-bedding tensor for conventional Scherk–Schwarz reductions was also calculated fromeleven dimensions using the “generalised vielbein postulate” which, as we discuss inthe conclusions, is connected to the algebra (1.3).The key point of this paper is to show that above conjecture also includes thesphere truncations. In contrast to the case of conventional geometry where it is afamous result that only S , S and S are parallelisable [33], we show that, withinan appropriate notion of generalised geometry, All spheres S d are generalised parallelisable. Furthermore we show for the round spheres they admit a frame with constant co-efficients X ABC encoding a SO ( d + 1) gauging. In the cases of S , S , S and S this generalised geometry (or an extension of it) encodes the appropriate ten- oreleven-dimensional supergravity. In particular we show that the frame algebra (1.3)reproduces the appropriate embedding tensor for the SO ( d +1) gauging of the reducedtheory, and the generalised Scherk–Schwarz deformations (1.4) match the standardscalar field ansatz for sphere consistent truncations [10, 35, 11, 34]. In the S case, weshould note that the tensor components of the parallelising generalised frame haverecently appeared in [31] building on the seminal work of [36, 9].The paper is organised as follows. In section 2 we define the GL + ( d + 1 , R )generalised geometry relevant to the S d generalised parallelisations. We define the– 3 –lobal generalised frame, show that (1.3) defines an so ( d +1) Lie algebra, and describethe generalised Scherk–Schwarz reduction of the scalar fields. Section 3 describes howthis structure encodes the classic sphere consistent truncations on S , S , S and S .As an application we derive the general scalar-field ansatz for the S truncation oftype IIB. Section 4 gives our conclusions. Let us start by showing how the round sphere S d with a d -form field strength F has avery natural interpretation as a parallelisation of a particular version of generalisedgeometry. This will provide the basic construction for each of our supergravityexamples. Consider a theory in d dimensions with metric g and d -form field strength F = d A ,satisfying the equations of motion R mn = 1 d − F g mn , F = d − R vol g , (2.1)where F = d ! F m ...m d F m ...m d . This admits a solution with a round sphere S d metricof radius R .We define various relevant geometrical objects on S d in Appendix A. Here wesimply note that, in terms of constrained coordinates δ ij y i y j = 1 with i, j = 1 , . . . , d +1, we can write the metric of radius R on S d asd s = R δ ij d y i d y j = R d s ( S d ) . (2.2)There are d + 1 conformal Killing vectors k i which satisfy k i ( y j ) = i k i d y j = δ ij − y i y j , g mn = R − δ ij k mi k nj , (2.3)with L k i g = − y i g . The rotation Killing vectors can be written as v ij = R − ( y i k j − y j k i ) , (2.4)with the SO ( d + 1) algebra under the Lie bracket[ v ij , v kl ] = R − ( δ ik v lj − δ il v kj − δ jk v li + δ jl v ki ) . (2.5)– 4 – .2 GL + ( d + 1 , R ) generalised geometry The original formulation of generalised geometry due to Hitchin and Gualtieri [13,14], considers structures on a generalised tangent space E ≃ T M ⊕ T ∗ M . There is anatural action of O ( d, d ) × R + on the corresponding frame bundle, and defining an O ( d ) × O ( d ) sub-structure, or equivalently a generalised metric G , captures the NSNSdegrees of freedom of type II theories. However, this is only one of family of possiblegeneralised geometries where one considers structures on different generalised tangentspaces [19–22]. These capture the bosonic degrees of freedom of the bosonic fieldsof other supergravity theories, in particular those of type II and eleven-dimensionalsupergravity.Since the sphere background has a d -form field strength it is natural to considera generalised geometry with a d ( d + 1)-dimensional generalised tangent space, E ≃ T M ⊕ Λ d − T ∗ M. (2.6)One can write generalised vectors V = v + λ ∈ E or, in components, as V M = (cid:18) v m λ m ...m d − (cid:19) . (2.7)As usual E is really defined as an extension0 −→ Λ d − T ∗ M −→ E −→ T M −→ . (2.8)If locally F = d A and A is patched by A ( i ) = A ( j ) + dΛ ( ij ) on U i ∩ U j (2.9)then the patching of E is given by v ( i ) + λ ( i ) = v ( j ) + λ ( j ) + i v j dΛ ( ij ) (2.10)where v ( i ) ∈ T U i and λ ( i ) ∈ Λ n − T ∗ U i . This means that, given a vector ˜ v , a form ˜ λ ,and a connection A then V = ˜ v + ˜ λ + i ˜ v A = e A ˜ V (2.11)is a section of E , where the last equation is just a definition of the “ A -shift” operatore A . In other words a choice of connection A defines an isomorphism between sections˜ V of T M ⊕ Λ d − T ∗ M and sections V of E .Given a pair of sections V = v + λ and W = w + µ the Dorfman or generalisedLie derivative is just the standard Dorfman bracket [13, 14] L V W = [ v, w ] + L v µ − i w d λ (2.12)One can also define the corresponding Courant bracket as the antisymmetrization q V, W y = ( L V W − L W V ) . (2.13)– 5 –his particular extension of the tangent space gives an interesting generalisedgeometry because there is a natural action of positive determinant transformations GL + ( d + 1 , R ) on E , where sections transform in the d ( d + 1)-dimensional bivectorrepresentation [22]. (The case of d = 4 was first considered in [37, 19, 29, 38].)Concretely, we write the generalised vector index M as an antisymmetric pair [ mn ]of GL + ( d + 1 , R ) indices, where m, n = 1 , . . . , d + 1, so that V M = V mn = ( V m,d +1 = v m ∈ T MV mn = λ mn ∈ Λ T M ⊗ det T ∗ M (2.14)where we are using the isomorphism Λ T M ⊗ det T ∗ M ≃ Λ d − T ∗ M between bivectordensities and ( d − λ mn = 1( d − ǫ mnp ...p d − λ p ...p d − , (2.15)where ǫ m ...m d is the totally antisymmetric symbol, with components taking the values ±
1. The GL + ( d + 1 , R ) Lie algebra acts as δV mn = R mp V pn + R np V mp , (2.16)and we can parameterise the Lie algebra element as R mn = (cid:18) r mn − r pp δ mn + cδ mn a m α n r pp + c (cid:19) . (2.17)where a m = 1( d − ǫ mp ...p d − a p ...p d − ∈ T M ⊗ det T ∗ M ≃ Λ d − T ∗ M,α m = 1( d − ǫ mp ...p d − α p ...p d − ∈ T ∗ M ⊗ det T M ≃ Λ d − T M. (2.18)In terms of v and λ we have δv m = cv m + r mn v n − d − α mn ...n d − λ n ...n d − ,δλ m ...m d − = cλ m ...m d − − ( d − r n [ m λ | n | m ...m d − ] + v n a nm ...m d − , (2.19)and we see that r mn parameterises the usual GL ( d, R ) action on tensors. We see thatthe corresponding adjoint bundle ad ˆ F decomposes asad ˆ F ≃ R ⊕ ( T M ⊗ T ∗ M ) ⊕ Λ d − T M ⊕ Λ d − T ∗ M (2.20)and is indeed ( d + 1) -dimensional. Note that a generates the “ A -shift” transforma-tion (2.11). Also setting c = ( d − d +1) r pp generates the SL ( d + 1 , R ) subgroup.– 6 –he partial derivative ∂ m naturally lives in the dual generalised vector space E ∗ ≃ T ∗ M ⊕ Λ d − T M as ∂ M = ∂ mn = ( ∂ m,d +1 = ∂ m ∂ mn = 0 . (2.21)One can write the generalised Lie derivative in GL + ( d + 1 , R ) form via the usualformula [23] ( L V W ) M = ( V · ∂ ) W M − ( ∂ × ad V ) M N W N , (2.22)where V · U denotes the contraction between elements of E and E ∗ , while U × ad V isthe projection from E ∗ ⊗ E onto the adjoint representation of Lie algebra gl ( d + 1 , R ).Concretely we have V · U = V M U M = V mn U mn , ( U × ad V ) mn = V mp U np − V pq U pq δ mn . (2.23)The form of L V given in (2.22) naturally extends to an action on any given GL + ( d +1 , R ) representation.As usual the bosonic degrees of freedom g and A , together with an extra overallscale factor ∆, parameterise a generalised metric G MN . Here G is invariant underan SO ( d + 1) ⊂ GL + ( d + 1 , R ) subgroup. Concretely, if V = e ∆ e A ˜ V , and usingthe definition (2.23) of the contraction V M U N , we have (cf. [37, 19, 29, 24] and seealso [39]) G ( V, V ) = G MN V M V N = g mn ˜ v m ˜ v n + d − g m n . . . g m d − n d − ˜ λ m ...m d − ˜ λ n ...n d − = V T · e − g mn + d − A mn ...n d − A nn ...n d − − A mn ...n d − − A nm ...m d − ( d − g m ...m d − ,n ...n d − ! · V (2.24)where g m ...m d − ,n ...n d − is short-hand for g [ m | n | . . . g m d − ] n d − antisymmetrised sepa-rately on the sets of m i and n i indices. The factor ∆ is related to warped compacti-fications in supergravity theories [23, 24] as we will see.Another way to view the generalised metric, and see more explicitly that it isinvariant under SO ( d +1), is to note that we can also consider generalised tensors thattransform in the fundamental ( d + 1)-dimensional representation of GL + ( d + 1 , R ).We define a ( d + 1)-dimensional bundle of weighted vectors and densities, as in [24]for the case d = 4, W ≃ (det T ∗ M ) / ⊗ (cid:0) T M ⊕ Λ d T M (cid:1) , (2.25) Throughout this paper whenever there is a an implied sum over p antisymmetric indices, asthe in V M U M in the first line of (2.23), our conventions are that the sum comes with a weight of1 /p !. – 7 –here sections K = q + t ∈ W can be labelled as K m = ( V m = q m ∈ (det T ∗ M ) / ⊗ T MV d +1 = t ∈ (det T ∗ M ) − / , (2.26)and we are using the isomorphism (det T ∗ M ) / ⊗ Λ d T M ≃ (det T ∗ M ) − / . Byconstruction E = Λ W . We then have an SO ( d + 1) metric given by G ( K, K ) = G mn K m K n = K T · e − ∆ √ g (cid:18) g mn g mn A n g np A p det g + g pq A p A q (cid:19) · K, (2.27)where A m is the vector-density equivalent to A m ...m d − defined in (2.18). One thenhas G ( V, V ) = G mp G nq V mn V pq . (2.28)giving the generalised metric on E .Just as for Einstein gravity we can always introduce a local orthonormal frame { ˆ E A } for G . Recall that E transforms as a bivector under GL + ( d + 1 , R ). Thus theframe also transforms as a two-form under SO ( d + 1) and so is naturally labelledby an antisymmetric pair of SO ( d + 1) vectors indices, and so we write the basisgeneralised vectors as { ˆ E ij } with i, j = 1 , . . . , d + 1. By definition, we have theorthonormal condition G ( ˆ E ij , ˆ E kl ) = δ ik δ jl − δ il δ jk . (2.29)Given the isomorphism (2.6) one can define a sub-class of orthonormal frames thattransform under an SO ( d ) subgroup of SO ( d + 1) and can be written in terms ofthe conventional orthonormal frame ˆ e a , and their dual one-forms e a , defined by themetric g . These are called “split frames” in [15, 23], and here are given byˆ E ij = ( ˆ E a,d +1 = e ∆ (ˆ e a + i ˆ e a A )ˆ E ab = d − e ∆ ǫ abc ...c d − e c ∧ · · · ∧ e c d − . (2.30)Note that, as described in [24] in the case of d = 4, one can always introduce acorresponding frame ˆ E i on W such that ˆ E ij = ˆ E i ∧ ˆ E j . For the split frame, thecorresponding (dual) frame { E i } ∈ W ∗ is given by E i = ( E a = g − / e − ∆ / ( e a − e a ∧ A ) E d +1 = g − / e − ∆ / vol g . (2.31)One can then write the generalised metric G mn in (2.27) in terms of the dual frame E i as G mn = δ ij E im E jn . (2.32)– 8 –t is important to note that any local rotation of the frameˆ E ′ ij = U ik U jl ˆ E kl , (2.33)where U ∈ SO ( d + 1), gives an equally good generalised orthonormal frame. Notethat U and − U actually generate the same transformation. Thus, when d is odd,the local group defined by the generalised metric is actually SO ( d + 1) / Z . In conventional geometry a parallelisable space is one that admits a global frame, thatis, where each basis vector ˆ e a is a globally defined smooth vector field. Topologicallyit means that the tangent space T M is trivial. It is a famous result due to Bott andMilner and Kervaire [33] that the only parallelisable spheres are S , S and S . Herewe show, by explicit construction, that by contrast every sphere S d is “generalisedparallelisable”.Generalised parallelisability means that the GL + ( d + 1 , R ) generalised vectorbundle (2.6) admits a global generalised frame and hence is trivial. On the spherewith flux F = d A , we define the global frame asˆ E ij = v ij + σ ij + i v ij A (2.34)where v ij are the SO ( d + 1) Killing vectors on S d given in (2.4) and σ ij = ∗ (cid:0) R d y i ∧ d y j (cid:1) = R d − ( d − ǫ ijk ...k d − y k d y k ∧ · · · ∧ d y k d − , (2.35)where the functions y i are the constrained coordinates δ ij y i y j = 1. To see that theframe is globally defined note that v ij = 0 when y i = y j = 0d y i ∧ d y j = 0 when y i + y j = 1 (2.36)so, while the vector and form parts can separately vanish, each combination ˆ E ij isalways non-zero. By construction, they are globally defined sections of E . Further-more, from (2.24) we have G ( ˆ E ij , ˆ E kl ) = v ij · v kl + σ ij · σ kl = δ ik δ jl − δ il δ jk (2.37)where we have used (A.10). We see that the frame is orthonormal with respect tothe generalised metric on the round sphere. Note the corresponding globally defineddual frame E i is given by E i = g − / (cid:0) R d y i + y i vol g − R d y i ∧ A (cid:1) . (2.38)which is clearly globally defined and non-vanishing since d y i = 0 when y i = 1.– 9 –e can also calculate the analogue of the Lie bracket algebra of ˆ E ij by calculatingthe generalised Lie derivatives. One finds L ˆ E ij ˆ E kl = [ v ij , v kl ] + L v ij ( σ kl + i v kl A ) − i v kl d (cid:0) σ ij + i v ij A (cid:1) = [ v ij , v kl ] + L v ij σ kl + i [ v ij ,v kl ] A − i v kl (cid:0) d σ ij − i v ij F (cid:1) = [ v ij , v kl ] + L v ij σ kl + i [ v ij ,v kl ] A, (2.39)where in going from the second to the third line we have used F = R − ( d −
1) vol g and the identity (A.11). Thus by (2.5) and (A.9) we have L ˆ E ij ˆ E kl = q ˆ E ij , ˆ E kl y = R − (cid:0) δ ik ˆ E lj − δ il ˆ E kj − δ jk ˆ E li + δ jl ˆ E ki (cid:1) . (2.40)We see that the generalised Lie derivative algebra of the frame is simply the Liealgebra so ( d + 1). SL ( d + 1 , R ) Scherk–Schwarz reduction on S d Recall that, given a conventional parallelisable manifold M , if the Lie bracket algebraof the frame ˆ e a [ˆ e a , ˆ e b ] = f abc ˆ e c (2.41)has constant f abc then the parallelisation defines a Lie algebra and we have a localgroup manifold: M is either a Lie group or a discrete, freely-acting quotient of aLie group. It is well-known that such spaces admit consistent truncations [1, 2],provided f abb = 0 [3]. The standard metric is given by a bilinear on the Lie algebra,for instance the Killing form, so g mn = δ ab ˆ e ma ˆ e ma . (2.42)The scalar fields of the truncated theory correspond to a Scherk–Schwarz [3] reduc-tion. One considers GL ( d, R ) rotations of the frame that are constant on M (thoughdepend on the coordinates x in the non-compact space)ˆ e ′ a = U ab ( x )ˆ e b , g ′ mn = H ab ( x )ˆ e ma ˆ e ma , (2.43)where the symmetric matrix H cd = δ ab U ac U bd parameterises the GL ( d, R ) /O ( d ) cosetspace of deformations.We have shown that the S d sphere is actually a direct generalised geometricanalogue of a local group manifold. It admits a globally defined orthonormal frame,and the generalised Lie derivative of the frame defines a Lie algebra so ( d + 1). Thusit is natural to consider a generalised Scherk–Schwarz reduction (1.4). The newgeneralised frame is given by ˆ E ′ ij = U ik ( x ) U j l ( x ) ˆ E kl (2.44)– 10 –here U ij ( x ) are GL ( d + 1 , R ) matrices, constant on M . The new inverse generalisedmetric is then given by G ′ MN = T ik T jl ˆ E Mij ˆ E Nkl . (2.45)where we define the symmetric object T kl = δ ij U ik U j l . In what follows we willactually only need to consider SL ( d + 1 , R ) transformations so we can take det T = 1.Thus T ij parameterises an SL ( d + 1 , R ) /SO ( d + 1) coset. Inverting (2.24), we findthe general form of the inverse metric, in terms of component fields g ′ , A ′ and warpfactor ∆ ′ , G ′ MN = e ′ g ′ mn g ′ mp A ′ pn ...n d − g ′ np A ′ pm ...m d − ( d − g ′ m ...m d − ,n ...n d − + A ′ pm ...m d − A ′ ′ pn ...n d − ! . (2.46)Comparing the two expressions givese ′ g ′ mn = T ik T jl v mij v nkl , e ′ ( A ′ − A ) m ...m d − = T ik T jl v ij, [ m σ kl,m ...m d − ] , (2.47)where the index on v ij in the second line is lowered using g ′ mn and A is the fixedpotential on the original undeformed S d . Since we are considering SL ( d + 1 , R )transformations we have det G ′ = deg G , implyinge d +1)∆ ′ (det g ′ ) − d − = (det g ) − d − . (2.48)The analysis of the metric then follows from that in [10, 35]. Using i v ij d y k = R − ( y i δ jk − y j δ ik ) and (A.13) we have (cid:0) T ik T jl v mij v nkl (cid:1) (cid:16) T − i ′ j ′ ∂ n y i ′ ∂ p y j ′ (cid:17) = R − (cid:0) T ik y k (cid:1) v mij ∂ p y j , = R − (cid:0) T ij y i y j (cid:1) δ mn . (2.49)Hence, using (A.15), we haved s ′ = R ( T kl y k y l ) / ( d − T − ij d y i d y j ,A ′ = − T kl y k y l ) R d − ( d − ǫ i ...i d +1 ( T i j y j ) y i d y i ∧ · · · ∧ d y i d +1 + A, e ′ = ( T kl y k y l ) ( d − / ( d − . (2.50)As we will see, for the cases of interest, this exactly agrees with the standard scalarfield ansatz for sphere consistent truncations [30, 40, 10, 35, 11, 34]. Note that the factor of comes from the normalisation (2.29). – 11 – Consistent truncations on spheres
We now discuss how the generalised parallelisability of S d relates to the classic super-gravity sphere solutions: the S near-horizon NS-fivebrane background, AdS × S in eleven-dimensional supergravity, AdS × S in type IIB, and AdS × S in eleven-dimensional supergravity.Each of these examples has a corresponding consistent truncation on the S d sphere to a seven-, five- or four-dimensional gauged supergravity theory. This hasbeen shown explicitly for S [9], S [10] and S [12] and for a subsector of S [11,34]. We will consider each example in turn, demonstrating how the generalisedgeometry encodes the embedding tensor and the scalar field ansatz for the consistenttruncation. In particular we give the general scalar ansatz for the S case. S and SO (3 , generalised geometry The solution of type II supergravity corresponding to the near-horizon limit of par-allel NS fivebranes has the form of a three-sphere times a linear dilaton background R , × R t × S [41] d s = d s ( R , ) + d t + R d s ( S ) ,H = 2 R − vol g ,φ = − t/R, (3.1)where R is the radius of the three-sphere.In terms of GL + (4 , R ) generalised geometry on the S the relevant generalisedtangent space is now E ≃ T M ⊕ T ∗ M, (3.2)and, since for d = 3 we can simply set c = 0 in the algebra (2.19) and restrict to an SL (4 , R ) action. The structure groups can be viewed as SL (4 , R ) ≃ SO (3 , , and SO (4) / Z ≃ SO (3) × SO (3) (3.3)where we have used the fact that for d odd the generalised metric is preserved by a SO ( d + 1) / Z group. We see that we have the original O ( d, d ) generalised geometryconsidered by Hitchin and Gualtieri [13, 14].The SO (4) generalised frame is simply ˆ E ij = v ij + σ ij − i v ij B (3.4)and the algebra (2.40) is the so (4) ≃ so (3) × so (3) Lie algebra. To see this in a basisthat is more conventional for O ( d, d ) generalised geometry, first introduce the usual Comparing with (2.34), we have identified B = − A to match the usual O ( d, d ) generalisedgeometry conventions. – 12 –eft- and right-invariant vector fields on S l + = l + i l = R − e − i ψ (cid:2) ∂ θ + i csc θ∂ φ − i cot θ∂ ψ (cid:3) , l = R − ∂ ψ ,r + = r + i r = R − e i φ (cid:2) ∂ θ + i cot θ∂ φ − i csc θ∂ ψ (cid:3) , r = R − ∂ φ , (3.5)with the corresponding left- and right-invariant one-forms λ + = R e − i ψ (d θ + i cos θ d φ ) , λ = R (d ψ + cos θ d φ ) ,ρ + = R e i φ (d θ + i sin θ d ψ ) , ρ = R (d φ + cos θ d ψ ) . (3.6)We also chose a gauge B = 2 R cos θ d φ ∧ d ψ. (3.7)Defining two SO (3) triplets ˆ E L ¯ a and ˆ E Ra as the anti-self-dual and self-dual combina-tions of ˆ E ij we haveˆ E L + = l + − λ + − i l + B = e − i ψ h (cid:0) R − ∂ θ − R d θ (cid:1) + i csc θ (cid:0) R − ∂ φ − R d φ (cid:1) − i cot θ (cid:0) R − ∂ ψ + R d ψ (cid:1) i , ˆ E L = l − λ − i l B = R − ∂ ψ − R d ψ, ˆ E R + = r + + ρ + − i r + B = e i φ h (cid:0) R − ∂ θ + R d θ (cid:1) + i cot θ (cid:0) R − ∂ φ − R d φ (cid:1) − i csc θ (cid:0) R − ∂ ψ + R d ψ (cid:1) i , ˆ E R = r + ρ − i r B = R − ∂ φ + R d φ. (3.8)These are the conventional left and right bases for the two SO ( d ) groups in generalisedgeometry (see for example [15] where they are labelled ˆ E − ¯ a and ˆ E + a ). They areorthonormal in the sense that, definingˆ E A = ˆ E Ra ˆ E L ¯ a ! , (3.9)we have η ( ˆ E A , ˆ E B ) = (cid:18) δ ab − δ ¯ a ¯ b (cid:19) ,G ( ˆ E A , ˆ E B ) = (cid:18) δ ab δ ¯ a ¯ b (cid:19) , (3.10)where η is the usual O (3 ,
3) metric, that is, if V = v + λ , η ( V, V ) = i v λ, (3.11)and G is the generalised metric (2.24) (with ∆ = 0). Under the generalised Liederivative the algebra reads L ˆ E L ¯ a ˆ E L ¯ b = q ˆ E L ¯ a , ˆ E L ¯ b y = R − ǫ ¯ a ¯ b ¯ c ˆ E L ¯ c ,L ˆ E Ra ˆ E Rb = q ˆ E Ra , ˆ E Rb y = R − ǫ abc ˆ E Rc ,L ˆ E L ¯ a ˆ E Ra = q ˆ E L ¯ a , ˆ E Ra y = 0 , (3.12)– 13 –nd we see the su (2) × su (2) algebra explicitly. It is known that there is a consistent truncation of type IIA supergravity on S [34, 12]giving a maximal SO (4) gauged supergravity in seven dimensions . Making a furtherconsistent truncation to the NSNS fields gives a half-maximal SO (4) gauged theory.The embedding tensor of the half-maximal gauged supergravity [42, 8] is a three-form X ABC where A = 1 , . . . SO (3 ,
3) vector index. If one raises oneindex with the O (3 ,
3) metric one can regard X ABC = ( X A ) BC as a set of so (3 , A . To define a gauged supergravity one requires thequadratic constraint [8] [ X A , X B ] = − X ABC X C . (3.13)In terms of the generalised geometry X is encoded in the frame algebra (1.3).The quadratic condition simply follows from the Leibniz property of the generalisedLie derivative and X can be interpreted as the generalised torsion of the uniquegeneralised derivative ˆ D satisfying ˆ D ˆ E A = 0 [23] (see also appendix C). This is againin complete analogy with the conventional geometrical structure of a local groupmanifold – there is a unique torsionful connection (the Weitzenb¨ock connection)satisfying ˆ ∇ ˆ e a = 0 such that the torsion of ˆ ∇ equals the structure constants ofthe Lie algebra. As in the conventional case, the generalised version ˆ D , discussedin [43, 44], can be defined if and only if the space is generalised parallelisable.For the S parallelisation, we see from (3.12) that X abc = R − ǫ abc , X ¯ a ¯ b ¯ c = R − ǫ ¯ a ¯ b ¯ c , (3.14)with all other components vanishing. In SL (4 , R ) indices the self-dual and anti-selfdual parts of X ABC correspond to X ij and X ′ ij and we have X ij = R − δ ij . Thisindeed matches the known embedding tensor for the SO (4) theory [46, 45].We can also identify the scalar fields of the truncated theory. Given the frameis always required to be orthonormal with respect to the SO ( d, d ) metric, that is η ( ˆ E ′ A , ˆ E ′ B ) = η AB , the scalar fields U AB in the generalised Scherk–Schwarz reduc-tion (1.4) parameterise an SO ( d, d ) / SO ( d ) × SO ( d ) coset. Specialising to the S case, and using GL + (4 , R ) indices we can follow the discussion of section 2.4. Wefind the form of the metric and B -field from (2.50)d s ′ = R T kl y k y l T − ij d y i d y j ,B ′ = R T kl y k y l ) ǫ i i i i ( T i j y j ) y i d y i ∧ d y i + B, e ′ = 1 . (3.15) Group manifolds always give consistent truncations [1], but viewing S as SU (2) would onlygive an SU (2) gauging, whereas here the full SO (4) group is gauged. – 14 –e see that the warp-factor ∆ ′ is trivial and the metric and B -field scalar dependenceon T matches exactly that for the S consistent truncation in [34, 12]. It is interesting to note that other parallelisations of E exist, and give differentgaugings and truncation ansatze on the same round S space. In particular, wecould choose a frame based solely on the left-invariant vectors and one-formsˆ E L ¯ a = l ¯ a − λ ¯ a − i l ¯ a B, ˆ E Ra = l a + λ a − i l a B. (3.16)The algebra now reads L ˆ E L ¯ a ˆ E L ¯ b = q ˆ E L ¯ a , ˆ E L ¯ b y = R − ǫ ¯ a ¯ b ¯ c ˆ E L ¯ c ,L ˆ E Ra ˆ E Rb = q ˆ E Ra , ˆ E Rb y = R − ǫ ab ¯ c ˆ E L ¯ c ,L ˆ E Ra ˆ E L ¯ a = q ˆ E Ra , ˆ E L ¯ b y = R − ǫ a ¯ b ¯ c ˆ E L ¯ c . (3.17)This is clearly a different gauging, not isomorphic under SO (3 ,
3) transformationsto the SO (3) × SO (3) gauging of the previous section, since the embedding tensor X MNP is now not self-dual. Instead it defines an SO (3) gauging [46, 45].This is really a convention flux compactification on a group manifold, where l a defines the conventional parallelisation. To match the usual description, we can fixa different convention for the generalised frame, taking the linear combinationsˆ E A = ( ˆ E a = (cid:0) ˆ E Ra + ˆ E La (cid:1) = l a − i l a B, ˆ˜ E a = (cid:0) ˆ E Ra − ˆ E La (cid:1) = λ a , (3.18)such that η takes the form η ( ˆ E A , ˆ E B ) = 12 (cid:18) δ ab δ ab (cid:19) . (3.19)The algebra then reads q ˆ E a , ˆ E b y = f abc ˆ E c + H abc ˜ E c , q ˆ E a , ˆ˜ E b y = − f acb ˆ˜ E c , q ˆ˜ E a , ˆ˜ E b y = 0 , (3.20)where f abc = R − ǫ abc , H abc = R − ǫ abc , (3.21)As usual, f abc characterises the Lie algebra of the group manifold (here su (2)) and H = H abc l a ∧ l b ∧ l c is the three-form flux [4, 7, 25].– 15 – .2 S and E generalised geometry We next consider the AdS × S solution [48, 47] of eleven-dimensional supergravityd s = d s (AdS ) + R d s ( S ) ,F = 3 R − vol g , (3.22)where R is the radius of the four-sphere and we are using the conventions of [23, 24].That this theory has a consistent truncation to seven dimensions has been provenby Nastase, Vaman and van Nieuwenhuizen [10].In terms of the GL + (5 , R ) generalised geometry on the S we have E ≃ T M ⊕ Λ T ∗ M. (3.23)However this is precisely the generalised (exceptional) geometry in four dimen-sions [19], where we identify the U-duality exceptional group and its maximallycompact subgroup E × R + ≃ GL + (5 , R ) and H ≃ SO (5) . (3.24)This geometry was discussed in the context of an extension of Double Field The-ory in [29, 38] and in the general context of exceptional generalised geometry andgeneralised curvatures in [23, 24].The embedding tensor X ABC in this case transforms in the + representationof SL (5 , R ) [49]. From the form of the frame algebra (2.40), one finds that the twocomponents are given by X ij = R − δ ij , X ijkl = 0 , (3.25)which reproduces the standard embedding tensor of maximal seven-dimensional SO (5)gauged supergravity [50]. The scalar field ansatz is given by (2.50) where A is athree-form. Again this agrees with the ansatz derived in [10]. S and E generalised geometry We next consider the AdS × S solution [51] of Type IIB supergravityd s = d s (AdS ) + R d s ( S ) ,F = 4 R − (vol g + vol AdS ) , (3.26)where R is the radius of the five-sphere, vol g is the volume form on S , vol AdS is thevolume form on AdS and F is the self-dual five-form RR flux. We are using theconventions of [52] for the type IIB supergravity.– 16 –f we keep the full degrees of freedom of the Type IIB theory, the GL + (6 , R )generalised geometry embeds in a larger (exceptional) E × R + generalised geome-try [19, 23]. This is summarised in appendix B, partly using results of Ashmore [53].One considers the 27-dimensional generalised tangent space [19] E ≃ T M ⊕ ( T ∗ M ⊕ T ∗ M ) ⊕ Λ T ∗ M ⊕ (Λ T ∗ M ⊕ Λ T ∗ M ) ,V = v + ρ α + λ + χ α . (3.27)where α labels a doublet of the IIB S-duality SL (2 , R ) group. There is an natu-ral action of E × R + on V ∈ E that preserves the symmetric top-form cubicinvariant [53] c ( V, V, V ) = i v λ ∧ λ + λ ∧ ρ α ∧ ρ α + ( i v ρ α ) χ α ∈ Λ T ∗ M, (3.28)where we lower SL (2 , R ) indices by u α = ǫ αβ u β . For V, V ′ ∈ E there is a generalisedLie derivative [23, 53], just as in (2.22) but now such that × ad projects onto the E × R + adjoint representation, L V V ′ = ( V · ∂ ) V ′ − ( ∂ × ad V ) V ′ = [ v, v ′ ] + L v ρ ′ α − i v ′ d ρ α + L v λ − i v ′ d λ + d ρ α ∧ ρ ′ α + L v χ ′ α − d λ ∧ ρ ′ α + d ρ α ∧ λ ′ . (3.29)This captures diffeomorphisms together with the type IIB gauge transformations ofNSNS and RR fields.There is also a generalised metric G which is invariant under the maximal com-pact subgroup H = USp (8) / Z ⊂ E × R + and unifies all the bosonic degreesof freedom along with the warp factor ∆ of the non-compactified space. The corre-sponding generalised orthonormal frame { ˆ E A } transforms in the representation of USp (8). For what follows we can actually use the decomposition under the subgroup SO (6) × SO (2) ≃ SU (4) / Z × SO (2) ⊂ USp (8) / Z , giving { ˆ E A } = { ˆ E ij } ∪ { ˆ E i ˆ α } , = ( , ) + ( , ) , (3.30)where i = 1 , . . . , α = 1 ,
2. The orthonormal condition reads G ( ˆ E ij , ˆ E kl ) = δ ik δ jl − δ il δ jk ,G ( ˆ E ij , ˆ E ak ) = 0 ,G ( ˆ E i ˆ α , ˆ E j ˆ β ) = δ ˆ α ˆ β δ ij . (3.31)Given the isomorphism (3.27) we can again define a sub-class of orthonormal framesthat transform under an SO (5) subgroup of SO (6) ⊂ USp (8) / Z . The corresponding“split” generalised frame { ˆ E A } , analogous to (2.30), can be written asˆ E ij = ( ˆ E a = ˆ E a , ˆ E ab = ǫ abc c c ˆ E c c c , , ˆ E i ˆ α = ( ˆ E a ˆ α = ˆ E a ˆ α , ˆ E α = ˆ E α , (3.32)– 17 –here ˆ E a = e ∆ (cid:0) ˆ e a − i ˆ e a B α − i ˆ e a A − B α ∧ i ˆ e a B α − B α ∧ i ˆ e a A − B α ∧ B β ∧ i ˆ e a B β (cid:1) , ˆ E a ˆ α = e ∆ e − φ/ (cid:0) ˆ f ˆ αα e a + B ˆ α ∧ e a − ˆ f ˆ αα A ∧ e a + B α ∧ B ˆ α ∧ e a (cid:1) , ˆ E abc = e ∆ e − φ (cid:0) e abc + B α ∧ e abc (cid:1) , ˆ E a ...a ˆ α = e ∆ e − φ/ ˆ f ˆ αα e a ...a . (3.33)We have the usual SL (2 , R ) frameˆ f ˆ αα = (cid:18) e φ/ C e φ/ − φ/ (cid:19) , (3.34)and define B ˆ α = ˆ f ˆ αα B α = ˆ f ˆ αα ǫ αβ B β and e a ...a n = e a ∧ · · · ∧ e a n . The split frameencodes the string-frame metric g , dilaton φ and warp factor ∆, while the NSNStwo-form is given by B and the RR form field potentials are C (0) = C , C (2) = B ,and C (4) = A . Note that the inverse generalised metric can be written as G − MN = δ AB ˆ E MA ˆ E NB = δ ik δ jl ˆ E Mij ˆ E Njk + δ ˆ α ˆ β δ ij ˆ E i M ˆ α ˆ E j N ˆ β . (3.35)Certain components of G − are given explicitly in (B.17).For the application to S we are interested in structures defined by the subgroups E × R + ⊃ GL + (6 , R ) × SL (2 , R ) ,H = USp (8) / Z ⊃ SU (4) / Z × SO (2) ≃ SO (6) × SO (2) , (3.36)where again SL (2 , R ) is the S-duality group. We find that the generalised tangentspace decomposes as E ≃ E (0) ⊕ E ( α ) , = ( , ) + ( ′ , ) , (3.37)where E (0) ≃ T M ⊕ Λ T ∗ M, E ( α ) ≃ T ∗ M ⊕ Λ T ∗ M. (3.38)Comparing with (2.25) we see that E ( α ) ≃ (det T ∗ M ) / ⊗ W ∗ . This means that itis a GL + (6 , R ) ≃ R + × SL (6 , R ) one-form weighted by a R + factor of (det T ∗ M ) − / .We now show that the S solution actually gives a parallelisation of the fulltangent space E . We define a frameˆ E A = ( ˆ E ij = v ij + σ ij − i v ij A for E (0) , ˆ E i ˆ α = ˆ f ˆ αα ( R d y i + y i vol g + R d y i ∧ A ) for E ( α ) , (3.39)where the SL (2 , R ) frame is simply (3.34) with constant dilaton φ and RR scalar C .Since the E (0) component is exactly of the type discussed in section 2 we just use the– 18 –rame (2.34) which we know is globally defined. For E ( a ) we note that d y i vanisheson y i = 1 so the frame is nowhere vanishing (and is essentially the dual of the ˆ E i frame on W ). It is easy to see that the parallelising frame (3.39) is orthonormal,satisfying (3.31), for the round sphere with flux background (3.26).We can again work out the algebra of the frame under the generalised Lie deriva-tive (3.29). Since ˆ E ai is closed this reduces to using the generalised Lie derivativefor the GL + (6 , R ) subgroup. We find L ˆ E ij ˆ E kl = R − (cid:0) δ ik ˆ E jl − δ il ˆ E jk − δ jk ˆ E il + δ jl ˆ E ik (cid:1) ,L ˆ E ij ˆ E k ˆ α = R − (cid:0) δ il δ kj ˆ E l ˆ α − δ jl δ ki ˆ E l ˆ α (cid:1) ,L ˆ E i ˆ α ˆ E jk = 0 ,L ˆ E i ˆ α ˆ E j ˆ β = 0 . (3.40)Note that unlike the previous examples we have L ˆ E A ˆ E B = q ˆ E A , ˆ E B y , (3.41)and (3.40) does not define a Lie algebra but rather a Leibniz algebra. S It is widely believed that there is a consistent truncation on S to an SO (6) maxi-mally supersymmetric d = 5 supergravity. The metric and five-form flux subsectorwas shown to be consistent in [11, 34], but otherwise there is no complete derivationof consistency. In the following we will show that generalised parallelisable struc-ture (3.39) reproduces the correct gauge structure and matches the known scalaransatz for g mn and A m ...m . Furthermore we will derive the full scalar ansatz includ-ing the remaining bosonic fields.The embedding tensor T ABC of five-dimensional maximally supersymmetric su-pergravity transforms in the representation of E [54]. Decomposing under SL (6 , R ) × SL (2 , R ) this splits as = ( , ) + ( , ) + ( ¯ , ) + (¯ , ) + ( , ) . (3.42)For the SO (6) gauging, only the ( , ) component is non-zero. Specifically, decom-posing the E index as A = { ii ′ , ˆ αi } , one has X ii ′ ,jj ′ kk ′ = X ij δ kk ′ i ′ j ′ − X i ′ j δ kk ′ ij ′ − X ij ′ δ kk ′ i ′ j + X i ′ j ′ δ kk ′ ij ,X j, ˆ γii ′ , ˆ β,k = (cid:0) X ik δ ji ′ − X i ′ k δ ji (cid:1) δ ˆ γ ˆ β , (3.43)with all other components vanishing. We see that the algebra (3.40) corresponds to X ij = R − δ ij in agreement with the standard SO (6) gauging embedding tensor [54].– 19 –he scalar fields in the truncation enter via the usual Scherk–Schwarz rotationˆ E ′ A ( x ) = U AB ( x ) ˆ E B , U = U ii ′ jj ′ U ˆ βii ′ ,j U i,jj ′ ˆ α U i, ˆ β ˆ α,j ! ∈ E . (3.44)Note that under GL + (6 , R ) × SL (2 , R ), given a generalised vector V A = ( V ii ′ , V ˆ αi )the cubic invariant is given by [53] c ( V, V, V ) =
12 16! ǫ i ...i V i i V i i V i i + V ij V ˆ αi V ˆ αj . (3.45)and U is defined as the transformation that leaves c invariant. Unlike the previouscases, we cannot easily parameterise the coset E / USp (8). However, compar-ing (B.17) for the split frame (3.32) we can read off expressions for the metric andpotentials e ′ g ′ mn = δ AB U Ajj ′ U Bkk ′ v mjj ′ v nkk ′ , e ′ B ′ αmn = δ AB U Ajj ′ U ˆ γB k ˆ f ˆ γα R v jj ′ [ m ∂ n ] y k , e ′ (cid:0) A ′ mnpq − B ′ αm [ n B ′ αpq ] − A mnpq (cid:1) = − δ AB U Ajj ′ U Bkk ′ v jj ′ m λ kk ′ npq , (3.46)where the spacetime index on v jj ′ in the last two expressions is lowered using g ′ mn .Note that totally antisymmetrizing the final expression eliminates the B ′ α terms.Comparing with (B.18) we also finde ′ (cid:0) e − φ ′ h ′ αβ g ′ mn − B ′ αmp g ′ pq B ′ βqn (cid:1) = δ AB U ˆ αA i U ˆ βB j ˆ f ˆ αα ˆ f ˆ ββ R ∂ m y i ∂ n y j , (3.47)which defines the new SL (2 , R ) metric h ′ αβ . These expressions are the direct ana-logues of those derived for the S truncation in [9, 36, 31].If one specialises to the case where U AB parameterises only the SL (6 , R ) / SO (6)subspace, that is take U ii ′ ja = U iajj ′ = 0 and U aibk = δ ba δ ji , we are back to the casediscussed in section 2. The two-form fields vanish, B α = 0, φ and C are alreadymoduli, and in additiond s ′ = R ( T kl y k y l ) / T − ij d y i d y j ,A ′ = 12( T kl y k y l ) R ǫ i ...i ( T i j y j ) y i d y i ∧ · · · ∧ d y i + A, e ′ = ( T kl y k y l ) / (3.48)which is in complete agreement with the ansatz of [11, 34]. S and E generalised geometry We next consider the AdS × S solution [48] of eleven-dimensional supergravityd s = d s (AdS ) + R d s ( S ) , ˜ F = 6 R − vol g , (3.49)– 20 –here R is the radius of the seven-sphere and ˜ F is the seven-form flux, that is theeleven-dimensional dual of the usual four-form. Here we are using the conventionsof [23, 24]. It is a classic result due to de Wit and Nicolai [9, 62] that this back-ground admits a consistent truncation to SO (8) gauged N = 8 supergravity in fourdimensions.Here we will give a new interpretation of this truncation in terms of generalisedgeometry. The generalised frame (3.55) described below has, in fact, already ap-peared in the work of [31] as has the form of the scalar ansatz [9, 30, 31]. However,the key new points are, first, to note that this frame is a parallelisation of the gener-alised tangent space and, second, that the SO (8) embedding tensor is encoded in theframe algebra under the generalised Lie derivative. This shows that the truncationactually falls within the class of generalised Scherk–Schwarz reductions.If we keep all the degrees of freedom of the eleven-dimensional supergravity,we are led to an E × R + (exceptional) generalised geometry. One considers thegeneralised tangent space [19, 20] E ≃ T M ⊕ Λ T ∗ M ⊕ Λ T ∗ M ⊕ ( T ∗ M ⊗ Λ T ∗ M ) ,V = v + ω + σ + τ, (3.50)which transforms as the representation under E × R + action, where a scalar k of weight k under R + is a section of (det T ∗ M ) k/ . Given V, V ′ ∈ E there is ageneralised Lie derivative [23] given by L V V ′ = ( V · ∂ ) V ′ − ( ∂ × ad V ) V ′ = L v v ′ + ( L v ω ′ − i v ′ d ω ) + ( L v σ ′ − i v ′ d σ − ω ′ ∧ d ω )+ ( L v τ ′ − jσ ′ ∧ d ω − jω ′ ∧ d σ ) , (3.51)which captures diffeomorphisms together with the gauge transformations of three-form and dual six-form gauge fields. There is also a generalised metric which isinvariant under the maximal compact subgroup H = SU (8) / Z and unifies all thebosonic degrees of freedom along with the warp factor ∆. The corresponding gen-eralised orthonormal frame { ˆ E A } transforms in the complex two-form C represen-tation of SU (8). For what follows we can actually use the decomposition under thesubgroup SO (8) ⊂ SU (8) / Z , giving { ˆ E A } = { ˆ E ij } ∪ { ˆ E ′ ij } , C = + . (3.52)The orthonormal condition reads G ( ˆ E ij , ˆ E kl ) = δ ik δ jl − δ il δ jk ,G ( ˆ E ij , ˆ E ′ kl ) = 0 ,G ( ˆ E ′ ij , ˆ E ′ kl ) = δ ik δ jl − δ il δ jk (3.53) The “ j -notation” for the T ∗ M ⊗ Λ T ∗ M component is described in [20, 23, 24]. – 21 –ote that the full SU (8) representation and its conjugate have the formˆ E αβ = − i γ ijαβ (cid:0) ˆ E ij − i ˆ E ′ ij (cid:1) , ¯ˆ E αβ = i γ ij αβ (cid:0) ˆ E ij + i ˆ E ′ ij (cid:1) , (3.54)where γ ijαβ are Spin (8) gamma matrices, α, β = 1 , . . . , SU (8) indices and weare matching the conventions of [23, 24]. Given the isomorphism (3.50) we can againdefine a sub-class of orthonormal frames that transform under an SO (7) subgroup of SO (8) ⊂ SU (8) / Z . The corresponding “split” generalised frame { ˆ E A } , analogousto (2.30), can be written asˆ E ij = ( ˆ E a = ˆ E a , ˆ E ab = ǫ abc ...c ˆ E c ...c , , ˆ E ′ ij = ( ˆ E ′ a = ˆ E a, ... , ˆ E ′ ab = ˆ E ab , , (3.55)where ˆ E a = e ∆ (cid:16) ˆ e a + i ˆ e a A + i ˆ e a ˜ A + A ∧ i ˆ e a A + jA ∧ i ˆ e a ˜ A + jA ∧ A ∧ i ˆ e a A (cid:17) , ˆ E ab = e ∆ (cid:16) e ab + A ∧ e ab − j ˜ A ∧ e ab + jA ∧ A ∧ e ab (cid:17) , ˆ E a ...a = e ∆ ( e a ...a + jA ∧ e a ...a ) , ˆ E a,a ...a = e ∆ e a ⊗ e a ...a , (3.56)where e ab = e a ∧ e b etc. and A and ˜ A are the three- and dual six-form poten-tials respectively. This particular form of E frame first appeared in an extended(4 + 56)-dimensional formulation of eleven-dimensional supergravity in [55]. It arosevia a non-linear realisation of E , following an embedding in E , in [56] and ingeneralised geometry in [23, 24]. It recently appeared in the context of extending theoriginal de Wit–Nicolai analysis of [36] in [31].For the current application to S we are interested in structures defined by thesubgroups E × R + ⊃ GL + (8 , R ) ,H ≃ SU (8) / Z ⊃ SO (8) / Z . (3.57)We find that the generalised tangent space decomposes under GL + (8 , R ) as E ≃ E (0) ⊕ E (1) , = + ′ , (3.58)where E (0) ≃ T M ⊕ Λ T ∗ M, E (1) ≃ Λ T ∗ M ⊕ (cid:0) T ∗ M ⊗ Λ T ∗ M (cid:1) . (3.59)– 22 –e now show that the S solution actually gives a parallelisation of the fulltangent space E . We define a frameˆ E A = ( ˆ E ij = v ij + σ ij + i v ij ˜ A for E (0) , ˆ E ′ ij = ω ij + τ ij − j ˜ A ∧ ω ij for E (1) , (3.60)where ω ij and τ ij are defined in (A.8). Note that ω ij = 0 when y i + y j = 1 whereas τ ij = 0 when y i = y j = 0 so each ˆ E ′ ij is non-vanishing. Furthermore, using the formof the generalised metric [23, 24] and (A.10) we see that the frame is orthonormal.Note that the SU (8) form (3.54) of this frame has already appeared in [31].We can again work out the frame algebra under the generalised Lie deriva-tive (3.51). Since ω ij is closed this reduces to using the generalised Lie derivative forthe GL + (8 , R ) subgroup. We find L ˆ E ij ˆ E kl = R − (cid:0) δ ik ˆ E lj − δ il ˆ E kj − δ jk ˆ E li + δ jl ˆ E ki (cid:1) ,L ˆ E ij ˆ E ′ kl = R − (cid:0) δ ki δ jp ˆ E ′ lp − δ li δ jp ˆ E ′ kp − δ kj δ ip ˆ E ′ lp + δ lj δ ip ˆ E ′ kp (cid:1) ,L ˆ E ′ ij ˆ E kl = 0 ,L ˆ E ′ ij ˆ E ′ kl = 0 . (3.61)Again, unlike the S and S examples, we have L ˆ E A ˆ E B = q ˆ E A , ˆ E B y , (3.62)and (3.61) defines a Leibniz algebra. To make the local SU (8) / Z symmetry moremanifest, and hence match more closely the de Wit–Nicolai formulation [9, 36, 31],we can use the combinations (3.54). The frame algebra then reads L ˆ E αβ ˆ E γδ = − i R − (cid:16) δ αγ ˆ E δβ − δ αδ ˆ E γβ − δ βγ ˆ E δα + δ βδ ˆ E γα (cid:17) ,L ˆ E αβ ¯ˆ E γδ = − i R − (cid:16) δ γα δ βǫ ¯ˆ E δǫ − δ γβ δ αǫ ¯ˆ E δǫ − δ δα δ βǫ ¯ˆ E γǫ + δ δβ δ αǫ ¯ˆ E γǫ (cid:17) . (3.63)Let us now connect to the consistent truncation. The embedding tensor T ABC of four-dimensional N = 8 supergravity transforms in the representation of E [57]. Decomposing under SL (8 , R ) this splits as = + ′ + + ′ . (3.64)For the SO (8) gauging, only the component is non-zero. Specifically, decomposingthe E index as as two pairs of indices ii ′ as in (3.60), we have X ii ′ jj ′ kk ′ = − X ii ′ kk ′ jj ′ = X ij δ kk ′ i ′ j ′ − X i ′ j δ kk ′ ij ′ − X ij ′ δ kk ′ i ′ j + X i ′ j ′ δ kk ′ ij (3.65)with all other components vanishing. We see that the algebra (3.61) corresponds to X ij = R − δ ij in agreement with the standard SO (8) gauging embedding tensor [57].– 23 –he scalar fields in the truncation enter via the usual Scherk–Schwarz rotation.In this case, this was already described in [9, 36, 31]. For completeness, let us includethem here. In our notation, one hasˆ E ′ A ( x ) = U AB ( x ) ˆ E B , U = (cid:18) U ii ′ jj ′ U ii ′ jj ′ U ii ′ jj ′′ U ii ′ jj ′ (cid:19) ∈ E (3.66)Following [31], comparing with the split frame (3.56) we can read off expressions forthe metric and potentials e ′ g ′ mn = δ AB U Aii ′ U Bjj ′ v mii ′ v njj ′ , e ′ A ′ mnp = δ AB U Aii ′ U B jj ′ v ii ′ [ m ω jj ′ np ] , e ′ (cid:16) ˜ A ′ m ...m − ˜ A m ...m +
12 5!2!3! A ′ m [ m m A ′ m m m ] (cid:17) = δ AB U Aii ′ U Bjj ′ v ii ′ m σ jj ′ m ...m , (3.67)where the index on v ii ′ is lowered using g ′ mn . Note that antisymmetrizing the last ex-pression eliminates the A ′ term. We also have, using the fact that | vol G | is unchangedby a E transformation [23] e ′ det g ′ = det g. (3.68)Finally, if one specialises to the case where U AB parameterises only the SL (8 , R ) / SO (8)subspace, that is take U ii ′ jj ′ = U ii ′ jj ′ = 0, we are back to the case discussed in sec-tion 2. The three-form A vanishes, andd s ′ = R ( T kl y k y l ) / T − ij d y i d y j , ˜ A ′ = − T kl y k y l ) R ǫ i ...i ( T i j y j ) y i d y i ∧ · · · ∧ d y i + ˜ A, e ′ = ( T kl y k y l ) / (3.69)matching the expressions in [35, 34]. This paper presents a unified description of maximally supersymmetric consistenttruncations in terms of generalised geometry. We have seen that there is a directanalogue of a local group manifold (or “twisted torus”), namely that the manifoldadmits what might be called a
Leibniz generalised parallelisation (or a “generalisedtwisted torus structure”). This means the generalised tangent space E admits a global generalised frame { ˆ E A } , such that under the generalised Lie derivative L ˆ E A ˆ E B = X ABC ˆ E C , (4.1)– 24 –ith constant X ABC . In general this defines a finite-dimensional Leibniz algebra.The existence of such a frame allows one to consider a supersymmetric generalisedScherk–Schwarz reduction and X ABC becomes the embedding tensor of the truncatedtheory. The key point of this paper was to show that the “exceptional” spherecompactifications are actually of this type. This relied on the demonstration thatall round spheres admit Leibniz generalised parallelisations. Viewed this way, theexceptional sphere truncations are no different from the conventional Scherk–Schwarzreductions on a local group manifold.A natural question to ask is how the unimodular condition f abb = 0 of conven-tional Scherk–Schwarz truncations [3] appears in the generalised context. As shownin [23], and summarised in appendix C, the embedding tensor X is equal to thetorsion of the generalised Weitzenb¨ock connection. However generically the torsionis a section of K ⊕ E ∗ [15, 23] where for O ( d, d ) × R + generalised geometry K ≃ Λ E and for E d ( d ) × R + generalised geometry the K representations are listed in [23].However, the embedding tensor lies only in the representation K [8] (provided thetheory has an action [58]) so there is a condition that the E ∗ component vanishes.This is the analogue of the unimodular condition and reads X BAB = 0 . (4.2)In appendix C we calculate X BAB for both the O ( d, d ) × R + (C.9) and E d ( d ) × R + (C.10)cases, given a conventional Scherk–Schwarz reduction with flux. If the dilaton φ andwarp factor ∆ are to be single-valued and bounded we see that X BAB = 0 ⇔ ( f abb = 0 , φ = const for O ( d, d ) × R + ,f abb = 0 , ∆ = const for E d ( d ) × R + . (4.3)Thus we indeed reproduce the standard unimodular condition for Scherk–Schwarzreductions. Note that (4.2) is also identically satisfied for the sphere truncations.In this paper, we have not proven that the truncations are consistent but onlyidentified the gauge structure in terms of the frame algebra and also the scalar fieldansatz. In general, one needs ansatze for the gauge fields and any other tensor fields,as well as the fermions. In the type IIB case, one already knows [11], for instance,that consistency requires the correct self-duality condition on the five-form flux,which we have not considered here. However, just as in conventional Scherk-Schwarzcompactifications, all these ansatze should follow simply from the existence of aglobal generalised frame. For example, the gauge fields appear as sections of E withgauge transformations generated by the generalised Lie derivative, so that [43, 59] A µ = A Aµ ˆ E A , δA µ = ∂ µ Λ − L A µ Λ = (cid:0) ∂ µ Λ C − X ABC A Aµ Λ B (cid:1) ˆ E C . (4.4)In general [59, 60], this will extend to a whole tensor hierarchy [61].– 25 –ecall that consistency of a conventional Scherk–Schwarz truncation is essentiallytrivial (see for example [2]). By expanding in the full set of left-invariant objects onecan never generate something outside the truncation, since such an object is bydefinition not left-invariant. Assuming the full ten- or eleven-dimensional theorycan be reformulated with the generalised geometry manifest, along the lines firstsuggested in [36], developed in [23, 24] for the internal space, and most recentlyformulated in full in [59, 60], one might expect that the proof of consistency forLeibniz generalised parallelisations is then equally straightforward.Though not the main point of this discussion, it is interesting to connect thegeneralised geometry of [23] to the internal “generalised vielbein postulate” (GVP)of [36, 62, 31], used to reformulate eleven-dimensional supergravity in a 4+7 split anddefined prior to any reference to a consistent truncation. The GVP is a differentialcondition on the generalised split frame { ˆ E A } (3.56) that has a form reminiscent ofthe usual relation between frame and coordinate expressions for a connection, namely ∂ m ˆ e na + Γ nmp ˆ e pa = ω mba ˆ e nb . Originally it was defined for only the vector componentof the frame ˆ E mA [36, 62]. This was then extended to all components in [31]. Thegeneralised geometry of [23, 24] gives a precise interpretation of the GVP. The GVPtakes the form ∇ m ˆ E MA + Ξ Mm N ˆ E NA = Ω mBA ˆ E MB , (4.5)where Ω m includes both Q m and P m defined in [31], Ξ has a restricted “triangular”form so that, for example Ξ nm P = 0, and ∇ is the Levi–Civita connection. This struc-ture matches (C.2) and (C.3) and so defines a particular generalised connection D GVP M .The key point is that only the first component D GVP m is non-zero, where generically,decomposing under GL (7 , R ), we have D M = ( D m , D m m , D m m ...m , D m,m ...m ). Infact, we can identify D GVP M directly: it is the standard lift D ∇ M of the Levi–Civita con-nection ∇ modified by flux-dependent terms. Explicitly, using the notation of [23],we have D GVP M V A = D ∇ M V A + Σ AM B V B (4.6)with Σ m = α ( ∂ m ∆) , Σ m a a a = βF ma a a , Σ am b = α ( ∂ m ∆) δ ab , Σ m a ...a = γ ˜ F ma ...a , (4.7)Note that D GVP M is not torsion-free. It is also an E -valued generalised connection,and, as such, in cannot directly act on spinors to define, for example, the super-symmetry variations. Instead in [36] the connection is split into SU (8) irreduciblerepresentations, and it is shown that the supersymmetry variations can be writtenin terms of these pieces, hence fixing the coefficients α , β and γ . As noted, D GVP M is defined using the GL (7 , R ) decomposition of E × R + since, for example, only To write the shift Σ m in terms matching the GVP one uses the standard transformationsbetween SL (8 , R ) and SU (8) indices as for example in eq. (B.30) of [20]. – 26 – GVP m is non-zero, and Ξ m has a particular form. In [23, 24] a different generalisedconnection was defined, that is, in some ways, more natural. By allowing all compo-nents of D M to be non-trivial, one can define a direct analogue of the Levi–Civitaconnection, namely a torsion-free, SU (8) generalised connection, with no need for adecomposition under GL (7 , R ). The internal supersymmetry transformations, andthe bosonic and fermionic equations of motion are then written directly using D M .Whether the appearance of this connection also extends to the full reformulation ofthe theory, including dependence on the additional four dimensions, is an interestingopen question. At least for the full O (10 , × R + formulation of type II theo-ries, we know that the analogous torsion-free connection is indeed the appropriateobject [16, 15].As a final point, it is obviously of importance to classify what spaces M admitsuitable generalised parallelisations. This would give a (possibly exhaustive) class ofmaximal gauged supergravities that appear as consistent truncations. Note first thatthe conditions of generalised parallelisability in general, and the existence of a Leib-niz generalised parallelisation in particular, are much weaker than the conventionalconditions of parallelisability and a local group manifold structure respectively, as isseen by the S and S examples. One condition [25] that can immediately be derivedfrom the existence of a Leibniz generalised parallelisation is that M is necessarily acoset space M = G/H , where g , the Lie algebra of G , is a subalgebra of the Leibnizalgebra (1.3). A general classification would thus address the old question of exactlywhich coset spaces admit consistent truncations [2]. Of particular interest is whetheror not the recently discovered family of four-dimensional, N = 8, SO (8) gaugings [63]appear as truncations within this class. Acknowledgments
We would like to thank Mariana Gra˜na, Michela Petrini and Chris Hull for helpfuldiscussions. We especially thank Anthony Ashmore for permission to use some of hisunpublished results on type IIB E × R + generalised geometry. K.L. is supportedby the National Research Foundation of Korea (NRF) grant funded by the Koreagovernment (MSIP) with the Grant No. 2005-0049409 (CQUeST). C. S-C. is sup-ported by the German Science Foundation (DFG) under the Collaborative ResearchCenter (SFB) 676 “Particles, Strings and the Early Universe”. D. W. is supported bythe STFC grant ST/J000353/1 and the EPSRC Programme Grant EP/K034456/1“New Geometric Structures from String Theory”.– 27 – The round sphere S d Consider Cartesian coordinates x i = ry i with δ ij y i y j = 1. The round metric g on S d of radius r = R is given by d s = R δ ij d y i d y j . (A.1)One has ∂∂x i = y i ∂∂r + k i r , (A.2)where k i are the conformal Killing vectors satisfying L k i g = − y i g. (A.3)In addition one has k i ( y j ) = i k i d y j = R − g ( k i , k j ) = R g − (d y i , d y j ) = δ ij − y i y j . (A.4)By considering d +1)! ǫ i ...i d +1 d x i ∧ d x i d +1 one can write the volume form on S d asvol g = R d d ! ǫ i ...i d +1 y i d y i ∧ · · · ∧ d y i d +1 . (A.5)We define the SO ( d + 1) Killing vectors v ij = R − ( y i k j − y j k i ) , (A.6)such that under the Lie bracket[ v ij , v kl ] = R − ( δ ik v lj − δ il v kj − δ jk v li + δ jl v ki ) , L v ij y k = R − ( y i δ jk − y j δ ik ) , L v ij d y k = R − (d y i δ jk − d y j δ ik ) . (A.7)It is also useful to define ω ij = R d y i ∧ d y j ,σ ij = ∗ ω ij = R d − ( d − ǫ ijk ...k d − y k d y k ∧ d y k d − ,τ ij = R ( y i d y j − y j d y i ) ⊗ vol g (A.8)Since y i and d y i transform in the fundamental representation under L v ij and vol g is invariant, we immediately have that all these tensors transform in the adjointrepresentation, that is, L v ij ω kl = R − ( δ ik ω lj − δ il ω kj − δ jk ω li + δ jl ω ki ) , L v ij σ kl = R − ( δ ik σ lj − δ il σ kj − δ jk σ li + δ jl σ ki ) , L v ij τ kl = R − ( δ ik τ lj − δ il τ kj − δ jk τ li + δ jl τ ki ) . (A.9)– 28 –e also have, contracting indices with the sphere metric, v ij · v kl := ( v ij ) m ( v kl ) m = y i y k δ jl − y j y k δ il − y i y l δ jk + y j y l δ ik ,ω ij · ω kl := ( ω ij ) mn ( ω kl ) mn = δ ik δ jl − δ il δ jk − ( y i y k δ jl − y j y k δ il − y i y l δ jk + y j y l δ ik ) ,σ ij · σ kl := d − ( σ ij ) m ...m d − ( σ kl ) m ...m d − = δ ik δ jl − δ il δ jk − ( y i y k δ jl − y j y k δ il − y i y l δ jk + y j y l δ ik ) ,τ ij · τ kl := d ! ( τ ij ) m,n ...n d ( τ kl ) m,n ...n d = y i y k δ jl − y j y k δ il − y i y l δ jk + y j y l δ ik . (A.10)Finally we note i v ij vol g = − R d − ( d − y i ǫ jk ...k d − y j ǫ ik ...k d ) y k d y k ∧ · · · ∧ d y k d = ( d − R d − ( d − y k ǫ ijk ...k d y [ k d y k ∧ · · · ∧ d y k d ] = R d − ( d − ǫ ijk ...k d − d y k ∧ · · · ∧ d y k d − = Rd − σ ij (A.11)where in the going to the second line we use y [ i ǫ i ...i d +2 ] = 0.We also use a couple of further identities. Defining the set of tensors A imn = v mij ∂ n y j we have A imn ∂ m y k = (cid:0) i v ij d y k (cid:1) ∂ n y j = R − (cid:0) y i δ kj − y j δ ki (cid:1) ∂ n y j = R − y i ∂ n y k . (A.12)Since d y i is an overcomplete basis for T ∗ M this implies that A i is proportional tothe identity matrix, namely, v mij ∂ n y j = (cid:0) R − y i (cid:1) δ mn . (A.13)Finally, suppose we have a metricd s ′ = R T − ij d y i d y j , (A.14)then [10, 35] one has det g ′ = ( T ij y i y j )det T det g. (A.15)This can be seen by considering the variation with respect to T ij δ log det g ′ = g ′ mn δg ′ mn = R δ ( T − ij ) g ′− (d y i , d y j ) . (A.16)– 29 –sing (A.13), one has g ′ mn = ( T ij y i y j ) − T kl T k ′ l ′ ( v kk ′ ) m ( v ll ′ ) n , (A.17)leading to δ log det g ′ = δ log ( T ij y i y j )det T , (A.18)which integrates to (A.15).
B Type IIB E generalised geometry In this appendix we summarise the main ingredients of E × R + generalised ge-ometry as applied to type IIB supergravity. The form of the generalised tangentspace was first given in [19]. The patching, generalised Lie derivative, and form ofthe split frame are given implicitly in [23] after applying the IIB decomposition de-scribed in the appendix C of that paper. Several of the explicit expressions givenhere were derived in unpublished work by Ashmore [53] and we are very grateful forthe permission to summarise them.One considers the 27-dimensional generalised tangent space [19] E ≃ T M ⊕ ( T ∗ M ⊕ T ∗ M ) ⊕ Λ T ∗ M ⊕ (Λ T ∗ M ⊕ Λ T ∗ M ) ,V = v + ρ α + λ + χ α . (B.1)This transforms in the representation of E × R + , with a weight one underthe R + -factor, where a scalar k of weight k is a section of (det T ∗ M ) k/ [23]. Thesplit of V above represents the decomposition under a SL (2 , R ) × GL (5 , R ) subgroupwhere SL (2 , R ) is the type IIB S-duality group. The symmetric E cubic invariantis given by [53] c ( V, V, V ) = i v λ ∧ λ + λ ∧ ρ α ∧ ρ α + ( i v ρ α ) χ α , (B.2)where we lower SL (2 , R ) indices by u α = ǫ αβ u β . Note that this is a five-form becauseof the weight of the generalised vector. There is a nilpotent subgroup of E thatacts as [53]e B α + A V = v − i v B α − i v A − B α ∧ i v B α − B α ∧ i v A − B α ∧ B β ∧ i v B β + ρ α + B α ∧ ρ α − A ∧ ρ α + B α ∧ B β ∧ ρ β + λ + B α ∧ λ + χ α , (B.3)where B α ∈ Λ T ∗ M and A ∈ Λ T ∗ M . As in (2.10) the generalised tangent space isreally patched by V ( i ) = e dˆΛ α ( ij ) +dΛ ( ij ) V ( j ) . (B.4)– 30 –f B α and A are two-form and four-form gauge potentials patched by B α ( i ) = B α ( j ) + d ˆΛ α ( ij ) ,A ( i ) = A ( j ) + dΛ ( ij ) + d ˆΛ ( ij ) α ∧ B α ( j ) , (B.5)the corresponding gauge-invariant field strengths are H α = d B α , F = d A − B α ∧ d B α . (B.6)As in (2.11) we can use the gauge potentials to define the isomorphism in (B.1) by V = e B α + A ˜ V , (B.7)where ˜ V is a sum a vector and p -forms (without additional patching). Given a pairof generalised vectors we have the generalised Lie derivative [23, 53] L V V ′ = ( V · ∂ ) V ′ − ( ∂ × ad V ) V ′ = [ v, v ′ ] + L v ρ ′ α − i v ′ d ρ α + L v λ − i v ′ d λ + d ρ α ∧ ρ ′ α + L v χ ′ α − d λ ∧ ρ ′ α + d ρ α ∧ λ ′ , (B.8)where × ad projects onto the E × R + adjoint.Let ˆ f ˆ αα be an SL (2 , R ) frame, and f ˆ αα the dual frame, which we can writeexplicitly in terms a parametrisation of SL (2 , R ) / SO (2) asˆ f ˆ αα = (cid:18) e φ/ C e φ/ − φ/ (cid:19) , f ˆ αα = (cid:18) e − φ/ − C e φ/ e φ/ (cid:19) . (B.9)If ˆ e a and e a are a conventional frame for T M and its dual, then we can define a splitframe by [23, 53]ˆ E a = e ∆ (cid:0) ˆ e a − i ˆ e a B α − i ˆ e a A − B α ∧ i ˆ e a B α − B α ∧ i ˆ e a A − B α ∧ B β ∧ i ˆ e a B β (cid:1) , ˆ E a ˆ α = e ∆ e − φ/ (cid:0) ˆ f ˆ αα e a + B ˆ α ∧ e a − ˆ f ˆ αα A ∧ e a + B α ∧ B ˆ α ∧ e a (cid:1) , ˆ E abc = e ∆ e − φ (cid:0) e abc + B α ∧ e abc (cid:1) , ˆ E a ...a ˆ α = e ∆ e − φ/ ˆ f ˆ αα e a ...a , (B.10)where B ˆ α = ˆ f ˆ αα B α = ˆ f ˆ αα ǫ αβ B β = ǫ ˆ α ˆ β f ˆ βα B α and e a ...a n = e a ∧ · · · ∧ e a n . Here thechoice of powers of the dilaton means that ˆ e a are vielbeins for a string-frame metric.The warp-factor ∆ is associated to compactifications with a string-frame metric ofthe form d s = e d s , + d s ( M ) (B.11)– 31 –here d s , is the metric in the non-compact five-dimensional space. Note that withthe SL (2 , R ) frame (B.9) we can define the complex three-form field strength chargedunder U (1) ≃ SO (2) G = − (cid:0) H ˆ1 + i H ˆ2 (cid:1) = − e − φ/ d B − ie φ/ (cid:0) d B − χ d B (cid:1) = i e φ/ (cid:0) τ d B − d B (cid:1) , (B.12)where τ = C + ie − φ . We then have the Bianchi identity for the five-form fieldstrength (B.6) d F = − H α ∧ H α = H ∧ H = i G ∧ G ∗ . (B.13)We see that our conventions for the gauge potentials and axion and dilaton matchthe standard definitions, as for example in [52]. The NSNS two-form is B , while theRR potentials are C (0) = C , C (2) = B and C (4) = A .Using the split frame we can define the generalised metric G by [23] G ( ˆ E A , ˆ E B ) = δ AB (B.14)where given { ˆ E A } = { ˆ E a , ˆ E a ˆ α , ˆ E abc , ˆ E a ...a ˆ α } we define (compatible with the conven-tions mentioned in footnote 1) δ a,b = δ ab , δ a a a ,b b b = 3! δ [ a | b | δ a | b | δ a ] b δ a,b ˆ α, ˆ β = δ ˆ α ˆ β δ ab δ a ...a , ˆ α b ...b ˆ β = 5! δ ˆ α ˆ β δ [ a | b | δ a | b | . . . δ a ] b , (B.15)with all other components vanishing. Equivalently we can define the inverse gener-alised metric as G − MN = δ AB ˆ E MA ˆ E NB . (B.16)In components, we note in particular that G − m,n = e g mn ,G − m,βn = e B β mn ,G − m,βn n n = − e (cid:0) A mn n n − B α [ n n B α mn ] (cid:1) . (B.17)and G − α,βm,n = e (cid:0) e − φ h αβ g mn − B αmp g pq B βqn (cid:1) , (B.18)where h αβ = δ ˆ α ˆ β ˆ f ˆ αα ˆ f ˆ ββ is the inverse SL (2 , R ) metric. Explicity one hase − φ h αβ = (cid:18) CC C + e − φ (cid:19) (B.19) This is not to be confused with the complex three-form G just defined. – 32 – Generalised connections and conventional Scherk–Schwarz
In this appendix we recall and expand slightly two of the results of [23]. First isthe relationship between the embedding tensor and the torsion of the generalisedWeitzenb¨ock connection. Second is the calculation of the embedding tensor for thespecific example of a conventional Scherk–Schwarz reduction on a local group man-ifold M . In [23] the calculation was for E d ( d ) × R + generalised geometry. Here wealso consider the O ( d, d ) × R + case.Recall that, given a conventional parallelisation, there is a unique connection ˆ ∇ m ,known as the Weitzenb¨ock connection, that preserves the frame, that is ˆ ∇ m ˆ e na = 0.However, generically ˆ ∇ m is not torsion-free, instead, the torsion T mnp is related tothe Lie algebra structure constants, T cab = − f abc , [ˆ e a , ˆ e b ] = f abc ˆ e c . (C.1)Let us now see how the analogous concepts arise in generalised geometry.A generalised connection [64, 15, 23] is a first-order linear differential operator D M which acts on generalised vectors as D M V N = ∂ M V N + Γ NM P V P . (C.2)Acting on a local frame { ˆ E A } one can define the analogue of the spin connection D M ˆ E NA = Ω BM A ˆ E NB . (C.3)The generalised one-forms Ω AB are Lie-algebra valued. If the corresponding groupis H we have an H -compatible generalised connection. If H ⊆ G , where G is thegeneralised structure group G (here E d ( d ) × R + or O ( d, d ) × R + ), we can also alwaysdefine the torsion T of the generalised connection as [15, 23] , given V ∈ E , T ( V ) = L DV − L V (C.4)where T ( V ) N P = V M T M N P is an element of the adjoint representation of G . Notethat in general the torsion lies in only particular irreducible representations of G [15,23] T ∈ K ⊕ E ∗ , (C.5)where for O ( d, d ) × R + we have E ≃ E ∗ and K = Λ E , while for E d ( d ) × R + one finds K transforms in the same representation as the embedding tensor, for example for E and for E . The key results of [15, 23] are first that There always exists a torsion-free, H -compatible generalised connection,where H is the maximally compact subgroup of G . Note that for O ( d, d ) × R + connections we are taking a slightly different convention from [15]for the ordering of the indices in T , so as to give a uniform treatment with the E d ( d ) × R + case. – 33 –nd second that, although this connection is not unique, there is a unique Ricci tensorwhich captures the bosonic equations of motion on the compactification space. (For O ( d, d ) this was first described using the DFT formalism in [16] and [17].) Further-more the internal contributions to the supersymmetry variations can be written interms of unique H -covariant projections of the connection, the generalised geometricanalogues of the Dirac operator [15, 23, 24].Just as in the conventional case, Ω is a global section of E ∗ if and only if { ˆ E A } is globally defined. If this is the case, given any generalised connection D , one canalways define a unique new connection ˆ D = D − Ω which satisfiesˆ D M ˆ E NA = 0 . (C.6)This is the generalised Weitzenb¨ock connection [43, 44]. As in the conventional case,the structure constants of the frame algebra are given by the generalised torsion (inframe indices) of the generalised Weitzenb¨ock connection [15, 23] X ABC = E C · (cid:0) L ˆ E A ˆ E B (cid:1) = − T AC B , (C.7)where { E A } is the dual generalised basis on E ∗ .Now suppose the generalised parallelisation arises from a conventional local-group manifold. Let ˆ e a be an invariant global frame for T M , for example the left-invariant vector fields. Let e a be the dual frame for T ∗ M . The split frame (3.18) for O ( d, d ) or (3.56) for E (more generally see eq. (3.19) of [15] and eq. (2.15) of [23]) isglobally defined, and gives a generalised parallelisation. Furthermore, we can identifythe generalised Weitzenb¨ock connection as the lift D ˆ ∇ M , as defined in [15, 23], of theconventional Weitzenb¨ock connection ˆ ∇ m . The corresponding torsion was calculatedin [15, 23]. One finds, for O ( d, d ) × R + , that the non-vanishing elements of the framealgebra are L Φ − ˆ E a ˆ E b = f abc ˆ E c + H abc ˆ E c − ( f acc + 2 ∂ a φ ) ˆ E b ,L Φ − ˆ E a ˆ E b = − f acb ˆ E c − ( f acc + 2 ∂ a φ ) ˆ E b ,L Φ − ˆ E a ˆ E b = f bca ˆ E c , (C.8)where H abc and ∂ a φ are the frame components of the flux and the derivative of the Note that for O ( d, d ) × R + generalised geometry [15], to incorporate the dilaton and O ( d, d )spinors correctly, one actually considers a “weighted” generalised tangent space ˜ E ≃ (det T ∗ M ) ⊗ ( T M ⊕ T ∗ M ) with a “conformal basis” { ˆ E A } (cf. (3.19)) satisfying η ( ˆ E A , ˆ E B ) = Φ η AB whereΦ ∈ det T ∗ M . The generalised torsion of the corresponding Weitzenb¨ock connection is then actuallygiven by X ABC = E C · (cid:0) L Φ − ˆ E A ˆ E B (cid:1) = − T AC B . – 34 –ilaton. For E d ( d ) × R + the non-vanishing elements are L ˆ E a ˆ E b = e ∆ h f abc ˆ E c + F abc c ˆ E c c + ˜ F abc ...c ˆ E c ...c + ( ∂ a ∆) ˆ E b − ( ∂ b ∆) ˆ E a i ,L ˆ E a ˆ E b b = e ∆ h − f ac [ b ˆ E | c | b ] + F ac ...c ˆ E b b c ...c − ˜ F acc ...c ˆ E c,b b c ...c + ( ∂ a ∆) ˆ E b b + 2( ∂ c ∆) δ a [ b ˆ E | c | b ] i ,L ˆ E a ˆ E b ...b = e ∆ h − f ac [ b ˆ E | c | b b b b ] + F acc c ˆ E c,b ...b c c + ( ∂ a ∆) ˆ E b ...b + 5( ∂ c ∆) δ a [ b ˆ E | c | b ...b ] i ,L ˆ E a ˆ E b,b ...b = e ∆ h − f acb ˆ E c,b ...b − f ac [ b ˆ E | b,c | b ...b ] + ( ∂ a ∆) ˆ E b,b ...b + ( ∂ c ∆) δ ab ˆ E c,b ...b + 7( ∂ c ∆) δ a [ b ˆ E | b,c | b ...b ] i ,L ˆ E a a ˆ E b = e ∆ h f bc [ a ˆ E | c | a ] + f c c [ a δ a ] b ˆ E c c − F c ...c δ [ a b ˆ E a c ...c ] − ∂ c ∆) δ [ cb ˆ E a a ] i ,L ˆ E a a ˆ E b b = e ∆ h f c c [ a ˆ E a ] b b c c − F c ...c ˆ E [ b ,b ] a a c ...c − ( ∂ c ∆) ˆ E ca a b b i ,L ˆ E a a ˆ E b ...b = e ∆ h f c c [ a ˆ E a ] ,b ...b c c + 2 f c c [ a ˆ E | c ,c | a ] b ...b − ∂ c ∆) ˆ E [ b ,b ...b ] ca a i ,L ˆ E a ...a ˆ E b = e ∆ h f bc [ a ˆ E | c | a ...a ] + 10 f c c [ a δ a b ˆ E a a a ] c c − ∂ c ∆) δ [ cb ˆ E a ...a ] i ,L ˆ E a ...a ˆ E b b = e ∆ h − f c c [ a ˆ E a ,a a a ] b b c c − f c c [ a ˆ E | c ,c | a ...a ] b b − ∂ c ∆) ˆ E [ b ,b ] ca ...a i , where again F abcd and ˜ F a ...a are the frame components of the fluxes and ∂ a ∆ isthe frame component of the derivative of the warp factor. 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