Covariantly constant forms on torsionful geometries from world-sheet and spacetime perspectives
aa r X i v : . [ h e p - t h ] O c t KCL-MTH-10-4
Covariantly constant forms on torsionful geometries fromworld-sheet and spacetime perspectives
P.S. Howe, G. Papadopoulos and V. Stojevic
Department of Mathematics, King’s College, London, UK
Abstract
The symmetries of two-dimensional supersymmetric sigma models on target spaces with covari-antly constant forms associated to special holonomy groups are analysed. It is shown that eachpair of such forms gives rise to a new one, called a Nijenhuis form, and that there may be furtherreductions of the structure group. In many cases of interest there are also covariantly constantone-forms which also give rise to symmetries. These geometries are of interest in the contextof heterotic supergravity solutions and the associated reductions are studied from a spacetimepoint of view via the Killing spinor equations. ontents H -flux 74 Worldsheet symmetry algebras 10 G = U ( m ) and G = SU ( m ); n = 2 m . . . . . . . . . . . . . . . . . . . . . . . . . 104.2 G = Sp ( k ); n = 4 k . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134.3 G = Sp ( k ) · Sp (1); n = 4 k . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144.4 Spin (7) and G . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195.2 Backgrounds with non-compact holonomy . . . . . . . . . . . . . . . . . . . . . . 225.2.1 SU(4) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 225.2.2 Sp(2), × Sp(1), Sp(1) and U(1) . . . . . . . . . . . . . . . . . . . . . . . . 245.2.3 Spin(7) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
It has been known for some time [1, 2, 3, 4, 5] that covariantly constant forms on a manifold M give rise to W-type symmetries in the context of two-dimensional (1 ,
0) and (1 ,
1) supersymmetricsigma models with target space M . At the same time, such forms arise naturally in heteroticstring backgrounds that preserve some of the spacetime supersymmetry, because the gravitinoKilling spinor equation (KSE) is a parallel transport equation with respect to a metric connection ∇ (+) with skew-symmetric torsion H . Therefore, all the forms that are constructed as bilinearsof the solutions of the gravitino KSEs are also ∇ (+) -parallel, and in turn, they generate W-type of symmetries (1.4) in the worldvolume theory. In this paper we examine the geometries1hich admit special holonomy forms, but which also have torsion, and investigate under whichcircumstances additional parallel forms exist and whether the structure group of M is reducedfurther due to their presence.Before going into the details, the following general comments are in order. The worldsheetand target space viewpoints are closely related, clearly, since imposing conformal invariancein the sigma model at the quantum level is what defines the stringy equations of motion [6].Nevertheless, different aspects of the analysis are more natural from one viewpoint than theother. On the sigma model side, the algebra of the W -symmetries is a powerful tool, whosestructure and closure tells us a lot about the geometry, already at the classical level. It is alsoof mathematical interest to work with sigma models on target spaces for which the conformalanomaly does not vanish, the most obvious example being a K¨ahler manifold that is not Calabi-Yau. The foremost disadvantages are that target space spinors are difficult to describe fromthe worldsheet perspective, and that the dilaton arises at the order α ′ , and is more difficultto study without going into the intricacies of quantisation. Therefore, studying the amount ofsupersymmetry preserved by a particular background is most easily done in terms of the KSEsand the field equations of heterotic supergravity.The target space ( M, g, H ) of (1 ,
1) and (1 ,
0) sigma models is an n -dimensional Riemannianmanifold ( M, g ) together with a closed three-form H , with H = db locally. There are two naturalmetric connections with torsion, Γ ( ± ) jik := Γ jik ± H jik , (1.1)where Γ is the Levi-Civita connection. The torsion tensors of the two connections are given by T ( ± ) ijk = ± H ijk . (1.2)Any covariantly constant ( l + 1)-form ω L , which we shall also see in the guise of a covariantlyconstant vector-valued l -form, L , i.e. ∇ (+) L = 0, defines a current j L = L iL D + X iL , (1.3)which is conserved when the equations of motion are satisfied, D − j L = 0 (or ∂ −− j L = 0 for (1,0)susy). (See appendix for conventions). The corresponding symmetry transformation is δ L X i = a L L iL D + X L (1.4)where the parameter a L has Lorentz weight − l , Grassmann parity ( − l and is independent ofthe minus coordinates. In the (1 ,
1) case one can have similar symmetries for forms which arecovariantly constant with respect to ∇ ( − ) , while in the (1 ,
0) case the L -type symmetries arerestricted to one sector. For l ≥ W -algebras. These have been studied, mainly in the torsion-free setting,in references [1, 2, 3, 4, 5, 7].For the (1,1) case in the absence of torsion the left and right forms are identified and it isnatural to consider those forms which are associated with irreducible holonomy, given by Berger’slist. The structure group is reduced from SO ( n ) to U ( m ) or SU ( m ), for n = 2 m , Sp ( k ) or Sp ( k ) · Sp (1), for n = 4 k , or to G or Spin (7) for two exceptional cases in n = 7 , H -flux is turned on, consideration of the worldsheet W-algebrashows that there may be additional covariantly constant forms which are not simply functionsof the original set of special holonomy forms; when this is the case, there will be further re-ductions of the structure group. Indeed, associated with any pair ( L, M ) of such forms thereis a covariantly constant form ˜ N ( L, M ) which we call the Nijenhuis form; it is related to theNijenhuis concomitant but is not, in general, the same object.The simplest case, and perhaps most important from a physical point of view, is when wehave an almost complex structure, I . If this is integrable, there is a second supersymmetry,and if we have a pair of them, I ( ± ) , we get a (2 ,
2) supersymmetric sigma model for whichthe target space is bi-Hermitian [8, 9]. Such geometries are now known to be equivalent togeneralised K¨ahler geometries [10, 11, 12, 13], and have been much studied in the recent literature[14, 15, 16, 17, 18, 19]. Here we consider geometries for which the I s are not complex. Thisproblem was investigated some time ago in [2] where it was shown that the Nijenhuis tensor N , which is equal to the Nijenhuis form ˜ N ( I, I ) in this case, is covariantly constant and hencedefines a new symmetry. Moreover, it appears that it generates an infinity of higher-ordersymmetries. The question of further reduction of the structure group was not addressed in [2],but this certainly does occur. It is not difficult to show that N is (3 ,
0) + (0 ,
3) with respect to ahermitian basis defined by I , so that, for example for m = 3, the structure group is automaticallyreduced to SU (3). For (1,1) models the most general case would involve two independent sets of left and right specialholonomy forms which give rise to two structure groups G ( ± ) which need not be isomorphic inprinciple. If they are, but the left and right forms are not the same, we have what might becalled a bi- G -structure. We shall focus for the most part on one sector and investigate whichsort of reductions can arise. Studying both sectors is relevant to type II string theories. Thebi- G -structures do not arise for the (1 ,
0) sigma model, which corresponds to the heterotic string.From the point of view of heterotic supergravity additional invariant one-forms arise when thedilatino KSE is not satisfied. The KSEs have been solved in generality in [20, 21]. The existenceof ∇ (+) -parallel spinors, and so solutions of the heterotic string gravitino KSE, requires thatthe holonomy of ∇ (+) reduces to a subgroup of the isotropy group G of the parallel spinors in Spin (9 , Spin (7) ⋉ R (1) , SU (4) ⋉ R (2) , Sp (2) ⋉ R (3) , × Sp (1) ⋉ R (4) ,Sp (1) ⋉ R (5) , U (1) ⋉ R (6) , R (8) , (1.5)and compact G (2) , SU (3)(4) , SU (2)(8) , { } (16) , (1.6)where in parenthesis is the number of invariant spinors. So, the solution of the gravitino KSEleads to the investigation of manifolds with G structures equipped with a compatible connectionwith skew-symmetric torsion.Apart from the parallel forms that can be constructed as parallel spinor bilinears, the spacetime,under certain conditions, may admit additional ∇ (+) -parallel forms. The presence of these forms An example of such a manifold is S equipped with the standard almost complex structure, which reducesthe structure to U (3). But since the almost complex structure is not integrable, the Nijenhuis tensor does notvanish reducing the structure further to SU (3). ∇ (+) to a subgroup of G . Such additional formshave been found in [21] by analyzing the dilatino KSE. In particular, the conditions that arisein the dilatino KSEs can be stated as the vanishing of certain forms, which we refer to as τ -forms. Now if the dilatino KSE is not satisfied, the τ -forms are no longer vanishing. However,it can be shown to be ∇ (+) -parallel subject to enforcing a Bianchi identity, dH = 0 and thefield equations of the theory. On one hand the existence of these forms breaks supersymmetrybecause the dilatino KSE is not satisfied, but on the other it leads to structure group reductionand thus the existence of additional parallel spinors. There are many examples of backgroundswhich are of this type. As example one can take the WZW models with constant dilaton.These are non-supersymmetric backgrounds, have 16 parallel spinors in the context of heteroticsupergravity, and solve the field equations.The backgrounds that solve the gravitino KSE are Lorentzian. To adopt the analysis to thecase of sigma models for which the target space is Euclidean, we shall extract the “Euclideancomponent” of the Lorentzian supersymmetric backgrounds. To do this, we make some simpli-fying assumptions on the structure of the Lorentzian manifolds. These assumptions are dictatedby the geometry of the Lorentzian supersymmetric backgrounds and depend on whether theisotropy group G of the parallel spinors is non-compact or compact. A more detail explanationwill be given in sections 5.1 and 5.2. Typically, we shall assume that the spacetime is metrically R − n, × X n , the fields are independent from the R − n, coordinates, and the 3-form flux hascomponents only along X n , where X n is the Euclidean component of the spacetime. In such acase, the holonomy of ∇ (+) reduces to a subgroup of Spin (7)[8] , SU (4)[8] , × Sp (1)[8] , Sp (1)[8] , U (1)[8] ,G [7] , SU (3)[6] , SU (2)[4] , Sp (2)[8] (1.7)where the number in the square brackets is the dimension n of X n .All the X n manifolds admit ∇ (+) -parallel forms which are the fundamental forms of the groups(1.7). As we have mentioned, these give rise to W-symmetries of the string world-volume action.We shall show that if the fields do not satisfy the dilatino KSE, then X n admits additional ∇ (+) -parallel forms subject to a Bianchi identity, dH = 0 and the field equations of the heteroticsupergravity. As a result, the string world-volume theory in these backgrounds admits additionalsymmetries, and the holonomy group of ∇ (+) on X n reduces to proper subgroup of those of (1.7).We shall investigate the patterns of reductions in each case. We shall demonstrate that in manyoccasions, the gravitino KSE admits additional parallel spinors which in turn trigger furtherreduction the holonomy.In section 2 we discuss the algebra of L -type symmetries due to special holonomy forms in ageneral setting. In section 3 we examine how new forms are generated both from the W-algebraand heterotic supergravity perspective. In section 4 we examine the W-algebras in detail goingthrough the special holonomy list, and in section 5 we go through the same list from the heteroticsupergravity perspective and also examine the structure group reduction in detail. We give someconcluding remarks in 6. The appendix summarises our notation and conventions. In this section we compute the commutator of two symmetry transformations of the type givenin (1.4), focusing for the most part on transformations of the same chirality. We shall deal with4eft symmetries (for which the parameters depend only on the plus coordinates) but there willbe no need to litter the formulae with pluses on the L -tensors. The general expression for thecommutator of symmetries based on special holonomy forms was computed in [3]; here we shallrewrite this so that we can identify the symmetries that arise in the presence of torsion. This wasbriefly outlined in [22]. Understanding the algebra of symmetries not stemming just from thespecial holonomy forms, but including for example the Nijenhuis or τ -forms, is best consideredcase by case. We delay the discussion of this to sections 3 and 4.The commutator is [ δ L , δ M ] X i = δ (1) LM X i + δ (2) LM X i + δ (3) LM X i , (2.1)where δ (1) LM X i = a M a L N ( L, M ) iLM D + X LM , (2.2) δ (2) LM X i = (cid:16) − ma M D + a L ( L · M ) jL ,iM + l ( − ( l +1)( m +1) a L D + a M ( L · M ) iL ,jM (cid:17) ×× D + X jL M , (2.3) δ (3) LM X i = − ilm ( − l a M a L ( L · M ) ( i | L | ,j ) M ∂ ++ X j D + X L M . (2.4)Here ( L · M ) iL ,jM := L ki [ L M k | j | M ] , (2.5)while N ( L, M ) denotes the Nijenhuis concomitant of L and M . We recall that a vector-valued l -form L defines a derivation, ι L , of degree l − p -forms (in Ω p ) to p + l − p ∋ ω ι L ω := pL iL ω iP dx LP ∈ Ω p + l − . (2.6)Since the graded commutator of two derivations is also a derivation we can define a new deriva-tion d L by d L := ι L d + ( − l dι L . (2.7)Clearly ι L generalises the interior product of a form with a vector field v while d L generalisesthe Lie derivative along v . Given two vector-valued forms we then find that[ d L , d M ] = d L d M − ( − lm d M d L = d N ( L,M ) . (2.8)This equation defines the Nijenhuis concomitant N ( L, M ). When L = M = I , an almostcomplex structure, N ( I, I ) is the usual Nijenhuis tensor. The explicit formula is N ( L, M ) i = (cid:0) L jL ∂ j M iM − M jM ∂ j L iL − lL ijL ∂ l M jM + mM ijM ∂ m L jL (cid:1) dx LM . (2.9)5n this formula we can replace the ordinary derivatives by the Levi-Civita covariant deriva-tive. For torsion-free sigma models on special holonomy target spaces, therefore, the Nijenhuisconcomitants vanish.There are at most three independent symmetries in the commutator but they do not corresponddirectly to the division exhibited in (2.1). To elucidate this structure we begin by writing theNijenhuis term in terms of the torsion, making use of the fact that both L and M are covariantlyconstant with respect to ∇ (+) . One finds N ( L, M ) iLM = − H ijk L jL M kM + mL jL H jm k M ikM − lM jM H jl k L ikL ++ lm H kl m ( L · M ) ( iL , k ) M . (2.10)The first line of the right-hand side is totally antisymmetric (when the i index is lowered), butis not covariantly constant in general. However, one can always add to it a term so that acovariantly constant ( l + m + 1)-form results. In order to see this and to simplify the remainingterms we use the following algebraic results, which can be proved for any of the special holonomyforms, ( L · M ) i [ L ,jM ] = ( − l +1 P ijL M + m g i [ j Q L M ] , ( L · M ) [ jL , | i | M ] = ( − l P ijL M + l g i [ j Q L M ] , ( L · M ) i [ L , | j | M ] + ( i ↔ j ) = g ij Q L M − ( l + m − g ( i [ l Q j ) L M ] . (2.11)The tensors P and Q are functions of the special holonomy forms and the metric that canbe found from the above equations; they are totally antisymmetric and covariantly constant;in particular cases they can vanish. Both of them can be used to define L -type symmetrytransformations, but in the commutator [ δ L , δ M ] Q is combined with the energy-momentumtensor. After some algebra one then finds that[ δ L , δ M ] X i = δ P X i + δ ˜ N X i + δ K X i , (2.12)where each term is now a symmetry by itself. The P transformation, which is of standard L -typehas parameter a P given by a P = ( − l +1 ma M Da L − ( − m lDa M a L . (2.13)The ˜ N transformation is also of this type; The Nijenhuis form ˜ N is given by˜ N iLM = − ( l + m + 1) (cid:18) H jk [ i L jL M kM ] + ( − l lm H [ il l Q L M ] (cid:19) . (2.14)It is not the Nijenhuis concomitant, since the latter is not totally antisymmetric in general, butit is constructed from it. It is not immediately obvious that ˜ N is covariantly constant. The A simple example is L = M = I , an almost complex structure, in which case P = 0 and Q = 1. a ˜ N is just a M a L .Finally, we consider the K transformation. If we define K i,K := g i [ k Q K ] , (2.15)where the multi-index K takes on l + m − Q ) that δ K X i = l + m − l + m − (cid:16) a K K j,iK ∂ ++ X j D + X K + i ( − k k ∇ (+)+ ( a K D + X K )+ ia K k ( H ijk Q K − k + 26 H [ ijk Q K ] ) D + X jK (cid:17) (2.16)is a symmetry of the action (A.2). In fact, the corresponding conserved quantity is the compositecurrent T Q . For the case in hand the parameter a K is a K = i ( − l +1 lm ( l + m − a M a L . (2.17)In summary, the commutator of two symmetries within the same sector determined by covari-antly constant special holonomy forms generically gives rise to three symmetries, two of whichagain involve covariantly constant forms and the third being generated by the product of theenergy-momentum and the current of another form. For (1 ,
1) sigma models the commutator ofa left and a right symmetry closes up to equation of motion terms [22]. H -flux In this section we describe the invariant forms that are potentially generated by the algebra of L -type symmetries when H is non-zero, as well as a set of invariant one-forms that arise fromthe target space perspective when the dilatino KSE is not satisfied. We refer to the formeras Nijenhuis forms, which we denote as ˜ N , and the latter as τ -forms. The τ forms are notgenerated by the special holonomy algebras and need to be introduced into the σ -model asadditional symmetries.For any two special holonomy forms, L, M , we define the Nijenhuis form ˜ N ( L, M ) by˜ N iLM = − ( l + m + 1) (cid:18) H jk [ i L jL M kM ] + ( − l lm H [ il l Q L M ] (cid:19) . (3.1)This is covariantly constant provided that the algebraic relations (2.11) are satisfied. With H turned on the algebra in general generates new forms and these in general obey relations differentto those in (2.11).To prove the covariant constancy of (3.1) one uses the Bianchi Identity, together with the factthat H is closed, to obtain 7 (+) p H ijk = 3 R (+)[ ij,k ] p . (3.2)We use the convention that the form indices on the curvature are the second pair. Applying ∇ (+) p to the first term in ˜ N we find ∇ (+) p H jk [ i L jL M kM ] = ( R (+)[ i | j,kp | + R (+) k [ i, | jp | + R (+) jk, [ i | p | ) L j L M kM ] . (3.3)The first two terms on the right vanish because L and M are invariant tensors under the holon-omy group and the first pair of indices on the curvature take their values in the correspondingLie algebra. Using the same fact, we see that the third term can be written R (+) jk, [ i | p | L j L M kM ] = lR (+) j [ l ,i | p | ) L j kL M kM ] = − lR (+) j [ l ,i | p | ( L · M ) j L ,M ] = lm − l +1 R (+)[ l l ,i | p | Q L M ] , (3.4)where, in the last line, we have used (2.11) and the invariance of P . From this it is easy to seethat ˜ N is covariantly constant as claimed.A further complication is that, as we shall discuss concretely below, some of the ˜ N forms caninduce a split in the tangent space, i.e. a reduction of the structure group to a product oftwo smaller groups. This implies that a covariantly constant almost-product structure R ij ispresent: R = 1 , ∇ (+) k R ij = 0 . (3.5)In general these structures are not integrable. Symmetries of (1 ,
1) models associated withcovariantly constant almost-product structures have been studied in detail in [23]. Integrabilityis equivalently expressed as the vanishing of the mixed parts of H with respect to the projectors P ij := 12 (cid:0) δ ij + R ij (cid:1) , Q ij := 12 (cid:0) δ ij − R ij (cid:1) . (3.6)Without assuming integrability one needs to carefully work out the combined algebra of thesuperconformal symmetries associated with non-integrable projectors together with L -type sym-metries. The complication is compounded by the fact that the projected version of the super-conformal transformation contains a non-linear piece involving the mixed part of H , and theeffect of this non-linearity needs to be carefully considered. Furthermore, without going throughthis analysis, we do not know the appropriate generalisation of (2.11). In this paper we willnot attempt to understand the general algebra, but will nevertheless be able to get a handleon many of the lower dimensional cases, because the analysis reduces to studying symmetriesrelated to covariantly constant one-forms.For this reason, and also in order to incorporate symmetries associated with the τ -forms (wediscuss these shortly), it will be useful to spell out the symmetry algebra involving a ∇ (+) -invariant vector v .The symmetry transformation is given by: 8 X i = a v v i , (3.7)and has the associated conserved current v i D + X i . The commutator with a superconformalsymmetry closes again to (3.7), and the commutator with an L -type symmetry yields two further L -type symmetries. The first of these is due to the l -form i v ω L , (3.8)while the second is due to the ( l + 1) Nijenhuis form˜ N ( v, L ) i i ··· i l +1 = v m H mp [ i L pi ··· i l +1 ] . (3.9)Clearly the former does not reduce the structure group, but the latter potentially does.The commutator of two vector-type symmetries (3.7) associated with v and w yields a Nijenhuisone-form. This object is essentially the Lie bracket with the index lowered, and can be writtenas ˜ N ( v, w ) i = H ijk v j w k , (3.10)using the covariant constancy of v and w . It follows that the structure group is potentiallyreduced further if v and w are linearly independent.The Lee form of a general form ω L is defined as θ L = − k L ⋆ ( ⋆dω L ∧ ω L ) . (3.11)The constants k L are determined by the requiring that the τ ω L one-form, defined as τ L := θ L − d Φ , (3.12)is covariantly constant [21] when we use the equations of motion of the heterotic string to thelowest order in α ′ , with Φ as the dilaton field and the gauge fields set to zero. For the particularexamples we consider, the constants k L are all listed in section 5.The equations of motion coming from the metric and b -field β -functions are : R ij − H ikl H klj + 2 ∇ i ∇ j Φ = 0 , (3.13) ∇ k H kij − ∇ k Φ) H kij = 0 . The equation of motion coming from the dilaton β -function is:4( ∇ Φ) − ∇ Φ − R + 112 H ijk H ijk + ( D − α ′ = 0 . (3.14)9t can be seen by contracting the first equation in (3.13) that for a constant dilaton, and when D = 10, the equations of motion can only be satisfied if also H = 0.Equations (3.13) are conveniently expressed in terms of R (+) ij , the Ricci tensor of the ∇ (+) connection, as: R (+) ij − ∇ (+) j ∇ i Φ = 0 . (3.15)For all the examples we consider the Lee form of ω L turns out to be the contraction of somecovariantly constant 4-form λ and H : ( θ L ) i ∝ λ ijkl H jkl . (3.16)The covariant constancy of τ L can be demonstrated straightforwardly from (3.2) and (3.15). Forexample, for the Lee form θ I associated with an almost-complex structure, we have( θ I ) m ∝ I [ ij I km ] H ijk . (3.17)It follows from (3.2) that ∇ (+) θ I is proportional to R (+) ij , provided that also I ij R (+) ijkm = 0 , (3.18)which is the requirement for the structure group to be in SU ( m ), rather than just U ( m ). It isthen obvious from (3.15) that ∇ (+) τ I = 0 imposes the metric and b -field equations of motion. In this section we discuss how the special holonomy algebras are deformed in the presence of H -flux by the Nijenhuis and τ forms. The full analysis of the structure group reduction is leftto section 5. G = U ( m ) and G = SU ( m ); n = 2 m The reduction of the holonomy group to U ( m ) is associated with an almost complex structure I .In this case, which has been studied for m = 3 in [2], one is dealing with the usual almost-complexNijenhuis form. In a hermitian frame basis the torsion can be decomposed into (3 ,
0) and (2 , ,
0) plus (0 ,
3) part of H . Combined with (3.2) this provides another wayof seeing that it is covariantly constant as the curvature tensor is pure on its Lie algebra indices,and therefore mixed when one of them is lowered. Although the Nijenhuis form is identicallycovariantly constant, it still implies a further reduction for the structure group.Further invariant forms arise if, in addition to I , we have a non-integrable almost-productstructure R (3.5) covariantly constant with respect to ∇ (+) . In [23] it is shown that, if we let { a, b, c } and { a ′ , b ′ , c ′ } denote indices associated with subspaces projected onto by P and Q ,10hen in addition to the purely (anti)-holomorphic components of H , which are related to thealmost-complex Nijenhuis form, also H abc ′ , H a ′ b ′ c , (4.1)as well as the complex conjugate components, are covariantly constant.In addition to an almost complex structure, the SU ( m ) holonomy group is associated withtwo real m -forms, L and b L which are the real and imaginary parts of a complex ( m,
0) form(with respect to a hermitian frame). In this case we have a number of possible Nijenhuis forms,˜ N ( I, L ) , ˜ N ( I, b L ) , ˜ N ( L, L ) , ˜ N ( b L, b L ) , ˜ N ( L, b L ) as well as ˜ N ( I, I ) which we will write as N . Apartfrom N it turns out that the only non-vanishing ones are ˜ N ( I, L ) and ˜ N ( I, b L ) and these aregiven in terms of N and L, b L . Therefore the only further reduction of the structure group is dueto the presence of the (3 ,
0) + (0 ,
3) form N .As an example we sketch the proof that ˜ N ( L, L ) vanishes for m even (it is identically zero for m odd). This is a (2 m − m − Q is proportional to I m − in this case so that the dual of the second term in ˜ N (equation(3.1)) is proportional to I ijkl H jkl . To evaluate the dual of the first term we use the fact that L isself-dual for m even to arrive at an expression of the form L p ...p m − ij L klp ...p m − H jkl ∝ I ijkl H jkl .A careful evaluation shows that the two terms cancel. The vanishing of ˜ N ( b L, b L ) and ˜ N ( L, b L )can be verified in a similar fashion.Now consider ˜ N ( I, L ). In this case Q = 0 since it involves the double contraction of I and L which are of different type with respect to the almost complex structure. It is again easier tolook at the dual, which is an l -form in this case (recall that m = l + 1). We find, for m even, ∗ ˜ N i ...i l = 1 m (cid:16) lH [ i jk b L jki ...i l ] + L ki ...i l I pq H pqk (cid:17) . (4.2)Because L and b L are both of type ( m,
0) + (0 , m ) it follows that this expression can be either( l,
0) or ( l − ,
1) or complex conjugates. It is not difficult to verify that the ( l,
0) part vanishesand this implies that only the (3 ,
0) + (0 ,
3) components of H contribute. But this is just N , sowe find ∗ ˜ N ...i l = l l + 1) N [ i jk b L jki ...i l ] . (4.3)Similar expressions can be derived for m odd and for ˜ N ( I, b L ). These forms, although non-zero,are generated from the original set together with N so that there is no further reduction of thestructure group.Next we need to consider Nijenhuis forms involving N . For m > N will induce a split in thetangent space. Due to the complications which were summarised in the context of (3.5), theanalysis in the next paragraphs applies only when the almost product structure associated withthis split is integrable.˜ N ( I, N ) ijkl ∝ H rs [ i I rj N skl ] is potentially non-vanishing. By going to a hermitian frame one cansee that the (3 ,
1) and (1 ,
3) parts vanish while the (2 ,
2) part involves only components of N .The only part we need consider is therefore the (4 ,
0) part. It is not difficult to see that it isproportional to ( H m [ ij H kl ] m ) , . On the other hand dH = 0 can be expressed as11 dH ) ijkl = ∇ (+)[ i H jkl ] + 32 H m [ ij H kl ] m = 0 . (4.4)In projecting out the (4 ,
0) component we eliminate the first term, since it is simply the covariantderivative of N , and the remainder of the expression implies that the (4 ,
0) component of ˜ N ( I, N )vanishes. Also, from the (2 ,
2) part of (4.4) we can see that the condition dH = 0 is incompatiblewith setting H = N , because then the (2 ,
2) component of (4.4) would be inconsistent unless H itself vanishes.We also need to consider the five-form ˜ N ( N, b N ), where b N jlm is given by I k [ j N lm ] k ( ˜ N ( N, N )vanishes identically).If the almost-product structure (3.5) is integrable, it follows straightforwardly that the (5 ,
0) +(0 ,
5) and (4 , ,
4) components of ˜ N ( N, b N ) are constructed from known covariantly constanttensors, and thus have no impact on the structure group. The former is zero for m = 3 and m = 4, and the latter is non-zero for m >
3. On the other hand, the (3 ,
2) + (2 ,
3) componentinvolves mixed parts of torsion and can potentially reduce the structure group. However, itturns out that at least for the m = 3 , SU ( m ) rather than U ( m ), taking θ I to be covariantly constant with respectto ∇ (+) implies that R (+) ij = 0. As discussed at the end of the previous section, these are thestringy equations of motion up to dilaton terms. Since θ I contracted with I is proportional tothe trace of H , the vanishing of θ I is equivalent to the primitivity condition I ij H ijk = 0 whichoften arises in the literature. The one-form I jk H ijk − I ik ∇ k Φ (4.5)is proportional to τ I contracted with I , and is covariantly constant under ∇ (+) provided thatthe stringy equations of motion (3.15) are satisfied, which is what we assume in the rest of thissection. It is not difficult to show that θ L is equal to θ I , so we potentially only have a singleadditional symmetry due to the τ one-forms.We now consider the particulars of the m = 3 , m = 3 it is obvious that the structure group is reduced to SU (3). Since N and b N arerespectively the real and complex parts of the holomorphic volume form, we have ˜ N ( N, b N ) = 0,and the algebra closes (as a W-algebra). The (4 ,
0) + (0 ,
4) part of (4.4) is now trivially zero. If H is primitive the algebraic part of the (3 ,
1) + (1 ,
3) component of (4.4) is zero, and followingfrom (3.2) the differential part implies relations between components of the curvature tensor.The symmetry due to τ I is not generated from the original SU (3) special holonomy symmetries,and must therefore be introduced separately. Let us introduce the hermitian basis of one-formsas e α = { e a , e } , a = { , } , together with their complex conjugates, which we choose so that τ I is entirely in the e / e directions. This reduces the structure group to SU (2).It follows from the general result (4.1) that, H ab , H ab , H a , H a , (4.6)12re all covariantly constant. The presence of the latter two components reduces the the structuregroup further to the trivial group. The covariant constancy of the components (4.6) can alsobe inferred by considering ˜ N ( I, τ I ), and the commutator between symmetries associated with τ I and τ I .For m = 4 we introduce a hermitian basis of forms e α , α = 1 , . . .
4, together with their complexconjugates e ¯ α . The covariant constancy of N implies that we can split an hermitian basis ofone-forms as e α = { e a , e } , a = { , , } , such that N abc = ǫ abc ; N ab = 0 . (4.7)Clearly the structure group is reduced to SU (3) × U (1). In SU (4) there are also covariantlyconstant (4 ,
0) and (0 ,
4) forms and so the structure group reduces to SU (3). In this case wecan study the symmetry algebra generated by the dual one-forms of N and b N which we call v and b v . It follows from (3.10) that the components H a , H a , (4.8)are covariantly constant. Their presence reduces the structure group to SU (2). Furthermore, itfollows from the covariant constancy of ˜ N ( I, v ) that H ab , H ab (4.9)are covariantly constant (actually, H ab = 0 due to the fact that N αβγ = H αβγ ), and the presenceof these components provides another way of breaking to SU (2). If the two ways of breaking to SU (2), (4.8) and (4.9), agree, the structure group remains SU (2), otherwise it reduces to thetrivial group. Again, the covariant constancy of the components (4.8) and (4.9) follows from thegeneral result (4.1).The (4 ,
0) and (0 ,
4) parts of (4.4), together with the vanishing of H ab , imply that the tracepart of H in the fourth direction vanishes: g ab H ab = g ab H ab = 0 . (4.10)This also turns out to be the condition for the commutators between symmetries associated v and b v , and symmetries associated with the four-forms L and b L to vanish.As in the SU (3) case, the symmetry due to τ I needs to be introduced separately, as it isnot generated from the original SU(4) special holonomy symmetries. θ I is proportional to I ij I kl H jkl , and it follows from (4.10) it can have no component in the fourth direction. Therefore,the gradient of the dilaton field determines whether τ I and ω v are linearly independent. Thecommutators (3.10) will then generate any additional invariant forms, the details of which dependon the direction in which the τ I form is pointing. G = Sp ( k ); n = 4 k Target spaces of this type could be called almost-complex HKT manifolds [24]. There is a setof Nijenhuis three-forms given by 13 rsijk = δ rs H ijk − H lm [ i ( I r ) lj ( I s ) mk ] , (4.11)where { I r } is a set of three almost complex structures obeying the algebra of the imaginaryquaternion units I r I s = − δ rs + ǫ rst I t . (4.12)These forms do not vanish. One way of understanding their content is to write a real vectorindex i = 1 . . . k as a pair i → ax , where now x = 1 . . . k and a = 1 ,
2. The latter index isacted on by the rigid Sp (1) while the former is acted on by Sp ( k ). In this notation a generalthree-form H can be written H ijk → H axbycz = H ( abc )[ xyz ] + ǫ ac H ′ byz,x + ǫ bc H ′ azx,y . (4.13)The H -tensor on the right has the indicated symmetries while H ′ axy,z is antisymmetric on thefirst two Sp ( k ) indices with the totally antisymmetric part being zero. In the Nijenhuis forms,this part of H drops out and so they are determined by H abcxyz . In detail, N rsaxbycz = ( σ r ) ( ad ( σ s ) be H c ) dexyz . (4.14)Adopting temporarily the notation I r = { I, J, K } , one can easily verify that there is only oneindependent covariantly constant object, since the covariant constancy of N ( I, I ) implies, viathe relations (4.12), the covariant constancy of all the other Nijenhuis forms, N ( I, J ), N ( J, J ),and so on.Starting from the θ forms, θ rs ∝ ⋆ ( ⋆dω ( r ∧ ω s ) ) , (4.15)where ω r is the two-form associated with I r , one can define six covariantly constant vectors τ rs .However, as five of these involve H abcxyz , only one of them is independent.For k = 1 the N tensors vanish, and the manifold is four dimensional hyper-K¨ahler with torsion.As Sp (1) = SU (2), this case overlaps with the m = 1 case of the previous subsection. If a τ formis present the structure group will reduce to the trivial group. For k = 2 the situation is similarto the m = 4 case in the last subsection, as Sp (2) ⊂ SU (4). It follows that with H abcxyz = 0the structure group reduces to at most SU (2). G = Sp ( k ) · Sp (1); n = 4 k Manifolds of this type can be called almost QKT spaces [25]. The only Nijenhuis form is theseven-form ˜ N ( L, L ) arising from the four-form L . In dimension 8 the dual can only be propor-tional to L ijkl H jkl . Since this object is not covariantly constant the constant of proportionalitymust be zero, as can be checked explicitly. Alternatively one can show that when n = 8, ˜ N ( L, L )can be written as ˜ N ( L, L ) ijklmrs = 12 H [ pqi L pjkl L qmrs ] , (4.16)14hich involves an antisymmetrisation over nine indices (an analogous proof is for the vanishingof the almost-complex Nijenhuis form in four dimensions, by writing it as an object involvingantisymmetrisation over five indices). For k > N ( L, L ) is not zero in general.
Spin (7) and G In Spin (7) there is a particular self-dual four-form φ whose corresponding Nijenhuis form ˜ N ( φ, φ )vanishes. The Lee form τ φ is proportional to φ ijkl H jkl , and if τ φ is non-vanishing the structuregroup reduces to G . It generates an additional symmetry of the type (3.7), which has a poten-tially non-vanishing commutator with the φ symmetry. This is an L -type symmetry generatedby a four-form. The algebra closes if this form vanishes, or if it is proportional to φ . Otherwisethe structure group is reduced further.In G there is a three-form ϕ and its dual four-form ∗ ϕ and the only non-zero Nijenhuis formis ˜ N ( ∗ ϕ, ∗ ϕ ). This a seven-form which is equal to the volume form multiplied by a constanttimes ϕ ijk H ijk . It is easy to see that this function is constant due to the structure of the g Lie algebra. The rest of the story is similar to the
Spin (7) case. The θ ϕ form is proportionalto ∗ ϕ ijkl H jkl , and if τ ϕ is non-zero the structure group reduces to SU (3). The potentially non-vanishing Nijenhuis forms are generated in the commutator of the of the related vector symmetrytransformation with the L -type symmetries of ϕ and ∗ ϕ . These are a three- and a four-form,respectively. The algebra closes if these are zero or proportional to the original G forms, andthe structure group is further reduced otherwise. Consider a solution of the gravitino KSE for which the isotropy group of the ∇ (+) -parallel spinorsis compact, see eqn (1.6). To identify the “Euclidean component” of the spacetime, we use theresults of [20, 21, 27] on the solution of KSEs. In particular, the gravitino KSE implies that theLorentzian spacetime admits 3, 4, 6 and 10 ∇ (+) -parallel, and so Killing, vectors, respectively.These are constructed as parallel spinor bi-linears. If, in addition, one assumes that the vectorspace spanned by these parallel vector fields closes under Lie brackets, which is not always impliedby the KSEs, and the infinitesimal action can be integrated to a free G -group action, then theLorentzian spacetime is a principal bundle with fibre a Lorentzian Lie group G and base space X n for n = 7 , , X n may depend on whether G is abelian or non-abelian. To extract the “Euclidean component”, we shall take G to be abelian ,the spacetime to be a metric product G × X n and the fields to depend only on the coordinatesof X n . Moreover, we shall assume that H has non-vanishing components only on X n . Underthese assumptions, X n is identified as the “Euclidean component” of the Lorentzian spacetime.With this definition, we exclude the parallelisable manifolds with non-vanishing torsion, ie allnon-abelian group manifold solutions of the heterotic string . Under these conditions, X n arespaces, of dimension n = 7 , ∇ (+) with skew-symmetric torsion andholonomy contained in G , SU (3) and SU (2), respectively. In what follows, we shall look at the We could give another definition of the “Euclidean component” to include those but the above definition willsuffice for the applications we consider. ∇ (+) -parallel forms on X n to those constructed from ∇ (+) -parallelspinor bi-linears. This is subject to imposing the Bianchi identity (3.2), dH = 0 and the fieldequations of the heterotic supergravity. These new forms in turn lead to the reduction of thestructure group to a subgroup of the isotropy group of ∇ (+) -parallel spinors. A similar reductionof the structure group for Lorentzian manifolds has also been noticed in [21] but its consequenceson the geometry of spacetime were not extensively explored.We shall show that each time that the holonomy of ∇ (+) reduces, the gravitino KSE admitsmore parallel spinors. These in turn give rise to more ∇ (+) -parallel forms which arise from theconditions on the geometry imposed by dilatino KSE. The new forms lead to further reductionof the holonomy of ∇ (+) . As a result, the structure of X n reduces in various patterns. Thegeometry of X n at each stage can be determined using the results of [20, 21, 27]. The algebraically independent fundamental forms on X are the Hermitian forms ω I (4) and ω J (4) which are associated with almost complex structures I (4) and J (4) with I (4) J (4) = − J (4) I (4) , ie X admits an almost hyper-complex structure ( I (4) , J (4) , I (4) J (4) ). The existence of a compatibleconnection with skew-symmetric torsion requires that N ( I (4) ) = N ( J (4) ) = 0 , i I (4) dω I (4) = i J (4) dω J (4) . (5.1)Moreover, the torsion is given as H = − i I (4) dω I (4) . (5.2)Therefore X is an HKT manifold. This is the full content of the gravitino KSE.The dilatino KSE also requires that τ I (4) = θ I (4) − d Φ , (5.3)vanishes, where θ I (4) = − ⋆ ( ⋆dω I (4) ∧ ω I (4) ) , (5.4)is the Lee form of ω L .Of course if τ I (4) vanishes, the SU (2) structure may not reduce. However let us assume that τ I (4) = 0. In such a case hol( ∇ (+) ) ⊆ SU (2), dH = 0, the identity (3.2) and the field equationsimply that ∇ (+) τ I (4) = 0 . (5.5)Complexifying the typical fibre of T X with respect to I (4) , SU (2) acts on it with the fundamen-tal 2-dimensional complex representation. Since in this representation, the isotropy group of avector in SU (2) is the identity, the structure reduces to { } . Therefore X is a group manifoldand so locally isometric to either S × S or T . The numerical subscript attached to the almost complex structures denotes the dimension of the associatedspace. Our form conventions are φ = k ! φ i ...i k dx i ∧ · · · ∧ dx i k and ( ⋆φ ) i ...i n − k = n − k )! ǫ j ...j k i ...i n − k φ j ...j k . .1.2 SU(3) The fundamental forms of X associated with an SU (3) structure are a Hermitian form ω I (6) and a (3,0)-form χ . The 6-dimensional manifolds with SU (3) structure and skew-symmetrictorsion have been extensively investigated, see eg [28, 29, 30, 31, 32, 33, 34]. For the manifold X with an SU (3) structure to admit a compatible connection with skew-symmetric torsion, N ( I (6) ) ijk = N ( I (6) ) [ ijk ] , θ I (6) = θ Re χ , (5.6)where θ I (6) = − ⋆ ( ⋆dω I (6) ∧ ω I (6) ) and θ Re χ = − ⋆ ( ⋆d Re χ ∧ Re χ ), ie the Nijenhuis tensor N ( I (6) ) must be skew symmetric in all three indices. Moreover, the torsion 3-form is completelydetermined in terms of the fundamental forms and the metric [35, 21] as H = − i I (6) dω I (6) − N ( I (6) ) = ⋆dω I (6) − ⋆ ( θ I (6) ∧ ω I (6) ) + N ( I (6) ) . (5.7)This concludes the analysis of the gravitino KSE.Turning to the dilatino KSE, one finds that N ( I (6) ) , τ I (6) , (5.8)must vanish. The gravitino and dilatino KSEs imply that X is a hermitian, conformally bal-anced manifold with hol( ∇ (+) ) ⊆ ( SU (3)). If in addition X is compact and dH = 0, then ithas been shown in [30] that it is Calabi-Yau. A non-compact example can be found in [36, 31].Step 1To investigate the pattern of reduction of the structure group, we shall follow the analysis inthe SU (2) case and assume that the conditions that arise in the analysis of the dilatino KSEare not imposed. Then hol( ∇ (+) ) ⊆ SU (3), dH = 0, the identity (3.2) and the field equationsimply that ∇ (+) N ( I (6) ) = 0 , ∇ (+) τ I (6) = 0 . (5.9)The first condition does not necessarily imply the reduction of the SU (3) holonomy. Instead, N ( I (6) ) is written as a linear combination of Re χ and Im χN ( I (6) ) = r Re χ + r Im χ , (5.10)where r , r are real constants. However, if τ I (6) = 0, the structure group reduces to SU (2).This is because SU (3) acts with the fundamental representation on the typical fibre of T X complexified with respect to R , and the isotropy group in SU (3) of a vector in this representationis SU (2).Step 2Now suppose that τ I (6) = 0 and so the connection with skew-symmetric torsion of X hasholonomy hol( ∇ (+) ) ⊆ SU (2). In such case, the gravitino KSE admits 4 additional ∇ (+) -parallelspinors and so 8 in total . The fundamental forms that can be constructed as ∇ (+) -parallelspinor bilinears are two 1-forms e a , a = 5 ,
6, as well as the Hermitian forms ω I (4) and ω J (4) of For the count the parallel spinors, we view the backgrounds as solutions of the gravitino KSE of heteroticsupergravity. I (4) and J (4) , where I = J = − × and I (4) J (4) = − J (4) I (4) . Moreover i a ω I (4) = i a ω J (4) = 0, a = 1 ,
2, where i a denotes inner derivation with respect to the vector field e a dual to 1-form e a . Compatibility with the SU (2) structure requires that one of the 1-forms,say e , must be θ I (6) − d Φ and the other is e = I (6) e .To give the conditions that arise from the gravitino KSE, we adapt a frame e = ( e a , e i ) on X ,where e a is identified with the first two parallel 1-forms and e i span the rest of the frame. Thenwe write ds = δ ab e a e b + d ˜ s , i, j = 1 , , , , a, b = 5 , H = 12 H abi e a ∧ e b ∧ e i + 12 H ija e i ∧ e j ∧ e a + ˜ H , (5.11)where d ˜ s := δ ij e i e j , ˜ H := 13! H ijk e i ∧ e j ∧ e k . (5.12)To analyze the gravitino KSE for the above background, we apply the results of [21], see appendixA, for the background R , × X with hol( ∇ (+) ) ⊆ SU (2). In particular, the conditions thatarise from the gravitino KSE can be written as( de a ) , , ij = − δ ab ( i I (4) ∇ b ω I (4) ) ij , ( de a ) ij ω ijI (4) = δ ab ( ∇ b ω J (4) ) ij ω ijK (4) , K (4) = I (4) J (4) N ( I (4) ) ijk = N ( J (4) ) ijk = 0 , (5.13)where the (2,0) and (0,2) part has been taken with respect to I (4) , and i a H = δ ab de b , ˜ H = − i I (4) ˜ dω I (4) = − i J (4) ˜ dω J (4) , (5.14)where ˜ d is the restriction of the exterior derivative along the e i directions. Observe that theconditions along the e i directions resemble those that we have found for the SU (2) case above.To proceed, we solve the dilatino KSE for a background R , × X with hol( ∇ (+) ) ⊆ SU (2) and8 Killing spinors. In particular, we find that ∂ a Φ , ( de a ) , , ij , de aij ω ijI (4) , [ e a , e b ] i , τ I (4) , (5.15)must vanish. Again using hol( ∇ (+) ) ⊆ SU (2), the identity (3.2), dH = 0, the field equations,and after some calculation, one can show that the tensors with components given in (5.15) areall ∇ (+) -parallel. The non-vanishing of some of the tensors in (5.15) does not necessarily lead tothe reduction of the holonomy. In particular, if ∂ a Φ , de aij ω ijI (4) = 0, the SU (2) holonomy does notreduce further. Similarly, if ( de a ) , , , ( de a ) , , = 0, one can set these tensors proportionalto ω J (4) and ω K (4) and the holonomy does not reduce. However if either [ e a , e b ] i = 0 or τ I (4) = 0,then X admits an additional linearly independent ∇ (+) -parallel 1-form. As in the SU (2) caseinvestigated previously, the structure reduces to { } and the spacetime is a group manifold.It is worth remarking that in the case that [ e a , e b ] i = 0, X admits two commuting ∇ (+) -parallelvector fields. This is because the only 2-dimensional metric Lie algebra with Euclidean signatureis R . If their action can be integrated to a T free group action, then X is a principal T bundle18ver a 4-dimensional manifold B . The geometry on B inherited from that of X depends onthe properties of de a . We have seen that i a ω I (4) = i a ω J (4) = 0 but( L a ω I r (4) ) ij = 2( de a ) k [ i ( ω I r (4) ) j ] k , r = 1 , , , (5.16)where I = I (4) , I = J (4) and I = K (4) = I (4) J (4) . Now if de a + ⋆de a = 2 f ar ω I r (4) , (5.17)then L a ω I r (4) = 2 f as ǫ srt ω I t (4) , (5.18)where f is constant. Thus in general, the hermitian forms are not invariant under the torusaction and so they do not decent as hermitian forms on B . There are two possibilities toconsider. One is that f = 0, ie ( de a ) , , = de aij ω ijI (4) = 0, then ω I (4) and ω J (4) decent ashermitian forms on B . Thus B admits an SU (2) structure compatible with a connection withskew-symmetric torsion, ie B is an HKT manifold. On the other hand, if f = 0, the integrabilitycondition of (5.18) and [ L a , L b ] = 0 imply that a linear combination of the three hermitian forms ω I r (4) is invariant under the torus action, see [27] for more details. As a result, B admits a U (2)structure compatible with a connection with skew-symmetric torsion, ie B is a KT manifold.Step 3Now suppose that hol( ∇ (+) ) ⊆ { } . In such a case, the spacetime is group manifold. The 6-dimensional Euclidean signature metric group manifolds are locally isometric to SU (2) × SU (2), SU (2) × T and T . Let ϕ and its dual ⋆ϕ be the fundamental forms of a 7-dimensional manifold X with a G structure. 7-dimensional manifolds with a G structure have been investigated in [37]. Suchmanifolds admit a compatible connection with skew-symmetric torsion [38], iff d ⋆ ϕ = θ ∧ ⋆ϕ , (5.19)where θ is the Lee form of ϕ θ = − ⋆ ( ⋆dϕ ∧ ϕ ) . (5.20)In such a case, the torsion is uniquely determined in terms of the fundamental forms as H = −
16 ( dϕ, ⋆ϕ ) ϕ + ⋆dϕ − ⋆ ( θ ∧ ϕ ) . (5.21)This is the full content of the gravitino KSE.The dilatino KSE requires that ( dϕ, ⋆ϕ ) , τ ϕ , (5.22)19ust vanish.Step 1Suppose that the conditions which arise from the dilatino Killing spinor are not imposed. Insuch a case, hol( ∇ (+) ) ⊆ G , the identity (3.2), dH = 0 and the field equations imply that ∇ (+) ( dϕ, ⋆ϕ ) = 0 , ∇ (+) τ ϕ = 0 . (5.23)Of course if ( dϕ, ⋆ϕ ) = 0, the holonomy does not reduce because it is a scalar. However, if τ ϕ = 0, the structure group of X reduces from G to SU (3).Step 2Suppose now that hol( ∇ (+) ) ⊆ SU (3) on X . In such a case, the gravitino KSE admits 2additional parallel spinors and so 4 in total. Moreover the parallel spinor bilinears on X area 1-form e , a hermitian form ω I (6) , i ω I (6) = 0 and I = − × , and a (3,0)-form χ , i χ = 0,where again i denotes the inner derivation with respect to the vector field e dual to the 1-form e . Therefore the fundamental forms are e and those of SU (3) case above in the directionsorthogonal to e .Adapting a frame e = ( e i , e ), i = 1 , . . . ,
6, the metric and 3-form are written as ds = ( e ) + d ˜ s , H = 12 H ij e ∧ e i ∧ e j + ˜ H ,d ˜ s = δ ij e i e j , ˜ H = 13! H ijk e i ∧ e j ∧ e k . (5.24)Applying the results of [21] for the manifold R , × X , hol( ∇ (+) ) ⊆ SU (3), where the metricand 3-form on X are given as above, the gravitino KSE requires that( de ) , , ij = − i I (6) ∇ ( ω I (6) ) ij , ( de ) ij ω ijI (6) = 16 ∇ Re χ ijk Im χ ijk . (5.25)These conditions are in addition to those in (5.6) along the directions orthogonal to e . Moreover, i H = de and ˜ H is given as in (5.7).Furthermore, the four ∇ (+) -parallel spinors also solve the dilatino KSE provided that ∂ Φ , N ( I (6) ) ijk , ( de ) ij ω ijI (6) , ( de ) , , ij , τ I (6) , (5.26)vanish. The structure reduces to SU (2) if either ( de ) , , = 0 or τ I (6) = 0. On the other handif ∂ Φ, N ( I (6) ), and ( de ) ij ( ω I (6) ) ij do not vanish, there is no further reduction of the SU (3)structure since these tensors are either scalars or can be chosen to be proportional to existing ∇ (+) -parallel forms as in (5.10).Now suppose that the infinitesimal action of e can be integrated to a U (1) free group action.In such a case X is a principal S fibration over a base space B . The geometric properties on B depend on de . As we have mentioned i ω I (6) = i χ = 0 but L ( ω I (6) ) ij = 2( de ) k [ i ( ω I (6) ) j ] k , L χ i i i = − de ) k [ i χ i i ] k . (5.27)20hus the hermitian form ω I (6) and the (3,0)-form χ may not decent on B . There are sev-eral cases to consider. One is that ( de ) , , = ( de ) ij ω ijI (6) = 0 which in turn implies that L ω I (6) = L χ = 0. Thus B admits a SU (3)-structure compatible connection with skew-symmetric torsion. Moreover, e is a principal bundle connection with curvature that obeys theHermitian-Einstein condition. One can also take ( de ) , , = 0 but ( de ) ij ω ijI (6) = 0. In thiscase, ω I (6) is invariant and so descents to a hermitian form on B but L χ = 0. Therefore B admits a U (3) structure compatible with a connection with skew-symmetric torsion but not an SU (3) structure. Finally, if both ( de ) , , , ( de ) ij ω ijI (6) = 0, then L ( ω I (6) ) , L χ = 0 and so B admits just an SO (6) structure.Step 3Suppose next that hol( ∇ (+) ) ⊆ SU (2). In such a case, X admits 8 ∇ (+) -parallel spinors. The ∇ (+) -parallel forms constructed from the spinor bilinears are three 1-forms e a , a = 5 , ,
7, andhermitian forms ω I (4) and ω J (4) , with i a ω I (4) = i a ω J (4) = 0, associated with the endomorphisms I (4) , J (4) , I = J = − × , I (4) J (4) = − J (4) I (4) . The analysis can proceed as for the reductionof the structure from SU (3) to SU (2) in section 5.1.2 step 2. The only difference is that thereis an additional parallel 1-form. The metric and 3-form can be written as ds = δ ab e a e b + δ ij e i e j , i, j = 1 , , , , a, b = 5 , , ,H = 13! H abc e a ∧ e b ∧ e c + 12 H abi e a ∧ e b ∧ e i + 12 H ija e i ∧ e j ∧ e a + ˜ H , (5.28)The rest of the formulae in section 5.1.2 step 2 on the conditions that arise from the gravitinoKSE still apply provided that the indices a, b = 5 , , H abc (5.29)must vanish. The holonomy of ∇ (+) further reduces provided some of the tensors in (5.15) donot vanish. The analysis is identical to that done for the SU (3) case.Assuming that the commutator the 3-vector field e a closes and their action can be integrated toa free group action, X is a either T or a SU (2) fibration over a 4-manifold B . The structureinherited on B from X depends on the conditions on de a and whether the Lie algebra of thevector fields is R or su (2). Setting again I = I (4) , I = J (4) and I = I (4) J (4) , and theself-dual part of de a , a = 5 , ,
7, as in (5.17), one again recovers (5.18) but now a = 5 , , R , then the analysis proceeds as in section5.1.2 step 2. B can either have an SU (2) or U (2) structure compatible with a connection withskew-symmetric torsion depending on whether or not f = 0. On the other hand if the Liealgebra of the vector fields is su (2), there is an additional possibility. This arises whenever f is anon-degenerate 3 × f is proportional to the identity. In such a case, B admits a SU (2) × SU (2)structure with anti-self-dual Weyl tensor, see also [27]. The previous two cases arise whenever f is a degenerate 3 × ∇ (+) ) ⊆ { } . In such a case, the spacetime is group manifold. The21uclidean signature metric group manifolds up to dimension 7 are locally isometric to SU (2) × SU (2) × U (1), SU (2) × T and T . The solution of the KSEs of backgrounds for which hol( ∇ (+) ) is non-compact and the reductionof the structure group of Lorentzian manifolds have been investigated in [20, 21]. These back-grounds always admit a ∇ (+) -parallel null vector field. To extract the “Euclidean component”,we shall separate the light-cone directions from the rest. For this, we shall assume that thespacetime is metrically R , × X , the fields are independent from the R , directions, and the3-form H has components only along the X directions. In such a case, the holonomy of the ∇ (+) reduces to a subgroup of Spin (7) , SU (4) , Sp (2) , × Sp (1) , Sp (1) , U (1) , (5.30)where we have excluded { } associated with the R case in (1.5). The number of ∇ (+) -parallelspinors of the holonomy groups (5.30) are as those in (1.5). The first three groups in (5.30) are inthe Berger list of holonomies for irreducible, simply connected Riemannian manifolds and act onthe typical fibre of T X with the spinor, fundamental, and spinor representations, respectively.The other three holonomy groups are new, and the way that act on T X will be described laterin detail, see also appendix B.Taking the holonomy of ∇ (+) as in (5.30), we shall again show that if one does not imposethe dilatino KSE, X admits new parallel forms. These in turn reduce the holonomy of ∇ (+) connection to subgroups of (5.30). This reduction leads to the existence of more parallel spinorson X which again give new parallel forms associated with the dilatino KSE. As a result thestructure of X reduces in patterns. One can determine the geometry of X at each stage byapplying the results of [20, 21, 27]. The fundamental forms of a manifold with an SU (4)-structure are a Hermitian form ω I , asso-ciated with an almost complex structure I (8) , and a (4,0)-form ψ . In order a manifold with an SU (4)-structure to admit a compatible connection with skew-symmetric torsion N ( I (8) ) ijk = N ( I (8) ) [ ijk ] , θ I (8) = θ Re ψ , (5.31)where the Lee forms are θ I (8) = − ⋆ ( ⋆dω I (8) ∧ ω I (8) ) , θ Re ψ = − ⋆ ( ⋆d Re ψ ∧ Re ψ ) , (5.32)and N ( I (8) ) is a (3,0)- and (0,3)-form. The torsion is completely determined in terms of themetric and the fundamental forms [35, 21] as H = − i I (8) dω I (8) − N ( I (8) ) = ⋆ ( dω I (8) ∧ ω I (8) ) − ⋆ ( θ I (8) ∧ ω I (8) ∧ ω I (8) ) + N ( I (8) ) . (5.33)The dilatino KSE imposes the conditions that N ( I (8) ) , τ I (8) , (5.34)22ust vanish. The first condition implies that X is a complex manifold and the second that X is conformally balanced.Step 1As in the previous cases, if the conditions that arise from the dilatino Killing spinor are notimposed, they give rise to new ∇ (+) -parallel forms on X . In particular, hol( ∇ (+) ) ⊆ SU (4),the Bianchi indentity (3.2), dH = 0 and the field equations imply that ∇ (+) N ( I (8) ) = 0 , ∇ (+) τ I (8) = 0 . (5.35)Now if the almost complex structure is not integrable dualising N ( I (8) ) with respect to Re ψ ,it gives rise to a ∇ (+) -parallel 1-form τ . Thus if either N ( I (8) ) = 0 or τ I (8) = 0, the structuregroup reduces to SU (3). If both are non-zero and linearly independent, then the structure groupreduces to SU (2).Step 2Now suppose that hol( ∇ (+) ) ⊆ SU (3). The spacetime admits two additional parallel spinors, ie4 in total. In turn, X admits two ∇ (+) -parallel 1-forms, e a , a = 7 ,
8, a Hermitian form ω I (6) and a (3,0)-form χ such that i a ω I (6) = i a χ = 0.The metric and H can be written as in (5.11) but now a, b = 7 , i, j, k = 1 , , . . . ,
6. Theconditions gravitino KSE imposes on X are like those we have found in the reduction of thestructure group from G to SU (3), ie those stated in equations (5.25) and (5.6). The onlydifference is that (5.25) holds for two 1-forms rather than one.The dilatino KSE implies that ∂ a Φ , N ( I (6) ) ijk , ( de a ) ij ω ijI (6) , ( de a ) , , ij , [ e a , e b ] i , τ I (6) , (5.36)must vanish. On the other hand if one of the last three tensors do not vanish, they give rise tonew ∇ (+) -parallel 1-forms on X which reduce the structure further to a subgroup of SU (2).Assuming that [ e a , e b ] i = 0 and the action of the vector fields can be integrated to a T freegroup action, X is a T fibration over a 6-dimensional manifold B . The geometry inherited on B from X depends on the properties of de a . The analysis is similar to the one we have doneat the end section 5.1.3 step 2 for the reduction from G to SU (3). The only difference is thatin the latter case, the fibre direction is one rather than two. Nevertheless the details remain thesame. B admits an SU (3) or U (3) structure compatible with a connection with skew-symmetrictorsion depending on whether ( de a ) ij ω ijI (6) = ( de a ) , , ij = 0 or ( de a ) ij ω ijI (6) = 0 , ( de a ) , , ij = 0,respectively. Otherwise it admits a SO (6) structure.Step 3Now suppose that hol( ∇ (+) ) ⊆ SU (2). The spacetime admits 8 ∇ (+) -parallel spinors in total. Insuch a case, X admits four ∇ (+) -parallel 1-forms, e a , a = 5 , , ,
8, and a Hermitian forms ω I (4) and ω J (4) , with I = J = − × and I (4) J (4) = − J (4) I (4) , such that i a ω I (4) = i a ω J (4) = 0. Thesolution to the gravitino KSE and the analysis that follows is similar to that we have explainedfor the reduction of G to SU (2) structure, and so we shall not give details. The only differenceis that in this case there are four parallel 1-forms rather than three. Moreover, if the action ofthe associated parallel vector fields can be integrated to a free group action, X is a principal23undle over a 4-dimensional manifold B with fibre either T or SU (2) × S . An analysis similarto that done in section 5.1.3 step 3 reveals that for both fibres B admits an either SU (2)or U (2) structure compatible with a connection with skew-symmetric torsion. If the fibre is SU (2) × S , then there is an additional case that arises. B admits an SU (2) × SU (2) structurewith anti-self-dual Weyl tensor.Furthermore, the structure can reduce to { } . In such a case X is a group manifold, and solocally isometric to T , T × SU (2), × SU (2) × T or SU (3). × Sp(1), Sp(1) and U(1)
To describe the geometry, we begin with the Sp (2) case. The condition hol( ∇ (+) ) ⊆ Sp (2)is equivalent to requiring that T X admits three endomorphisms I, J and K that satisfy thealgebra of imaginary unit quaternions, I = J = − , K (8) = I (8) J (8) . Moreover for each ofthe three almost complex structures, the torsion H can be written as (5.33) provided that theassociated Nijenhuis tensor is skew-symmetric in all indices. Since if I and J are parallel, so is K ,and since H must be the same for all almost complex structures, apart from the skew-symmetrycondition on the Nijenhuis tensors of I and J , one also requires that i I (8) dω I (8) + 2 N ( I (8) ) = i J (8) dω J (8) + 2 N ( J (8) ) . (5.37)This is the content of the gravitino KSE.The dilatino KSE implies that N ( I (8) ) , N ( J (8) ) , τ I (8) , τ J (8) , (5.38)must vanish. So X admits a hyper-complex structure, and so it is a conformally balanced HKTmanifold.To generalise the above discussion to × Sp (1), Sp (1), and U (1), we shall follow [26] and observethat in the Sp (2) case the tangent bundle of T X is a Cliff( R ) module, where Cliff( R ) is takenwith the negative definite inner product on R . The basis { i, j } of Cliff( R ) are representedby { I (8) , J (8) } , respectively, while K (8) corresponds to the even Clifford element k = ij . Forthe rest of the cases, × Sp (1) , Sp (1) and U (1), T X is a Cliff( R n ) module for n = 3 , T X which corresponds to an additional basis element of the Cliffordalgebra which anti-commutes with all the previous ones. Thus in each case T X admits theaction of n almost complex structures I r (8) , r = 1 , , . . . , n , which are all algebraically independentsuch that ( I r (8) ) = − I r (8) I s (8) = − I s (8) I r (8) for r = s . Of course T X admits the action of almostcomplex and almost product structures which can be constructed by taking products of the n basis elements. With these data, the conditions that arise from the gravitino KSE are i r dω r + 2 N ( I r (8) ) = i s dω s + 2 N ( I s (8) ) , r = s (5.39)and that N ( I r (8) ), r = 1 , . . . , n , is skew-symmetric in all indices.Similarly, the dilatino KSE implies that N ( I r (8) ) , τ I r (8) , r = 1 , . . . , n , (5.40)must vanish. We have shown in appendix B that if hol( ∇ (+) ) ⊆ × Sp (1) and two commutingcomplex structures are integrable, then X factorizes to a product X = X × X ′ with X and X ′ Sp (1) structure compatible with a connection with skew-symmetric torsion.Under the same assumptions if the holonomy of X is a subgroup of Sp (1) or U (1), then X isparallelisable.Before we examine the additional parallel forms that arise for those backgrounds, we shall firstinvestigate the way that the structure groups Sp (2), × Sp (1), Sp (1) and U (1) act on the typicalfibre of T X . Identifying the typical fibre of T X with H , Sp (2), represented with 2 × H from the left. Then the action of I (8) , J (8) and K (8) on thetypical fibre can be identified with the action of i, j and k , the quaternion basis, on H fromthe right. Clearly, this action commutes with that of Sp (2), ie I (8) , J (8) and K (8) are invariantunder Sp (2) as expected.Next the structure group × Sp (1) can be identified with diagonal subgroup of Sp (2). With thisidentification, the action of × Sp (1) on H commutes with the endomorphism Π : x ⊕ y → x ⊕ − y . Moreover Π commutes with all I (8) , J (8) and K . So if { I (8) , J (8) } is chosen as thebasis of Cliff( R ) associated with the Sp (2) case, then a basis for Cliff( R ) is { I , I , I } = { I (8) , J (8) , Π I (8) J (8) } .Similarly, Sp (1) structure group can be identified with the diagonal subgroup of × Sp (1). Insuch a case, the action of Sp (1) on H commutes with the endomorphism Σ : x ⊕ y → y ⊕ x .Moreover Σ commutes with I , I but anti-commutes with I . The additional basis elementof Cliff( R ) associated with Sp (1) can be chosen as I = Σ I (8) J (8) .It remains to investigate the action of U (1) structure group on H . If Sp (1) is identified with thequaternions of length one, then U (1) is the subgroup of Sp (1) spanned by the complex numbersof length one. It is clear then that the action of U (1) on H commutes with the action T ofimaginary unit i on H acting from the left. In addition, T commutes with I r (8) , r = 1 , . . . , T = −
1. Using this the additional basis element of Cliff( R ) associated with the U (1)holonomy group can be chosen as I = I I I I T . Observe that ( I ) = − I anticommutes with the other four basis elements.Step 1Having established the action of the structure groups on the typical fibre of T X , we can nowinvestigate their reduction in the cases that additional forms are ∇ (+) -parallel. To do thisobserve that X admits an SU (4) structure with respect to each almost complex structure I r (8) .Suppose that the dilatino KSE is not satisfied. In such a case, the Nijenhuis tensor N ( I r (8) ) maynot vanish. Each such Nijenhuis tensor is skew-symmetric in all three indices and (3,0)- and(0,3)- form with respect to the associated almost complex structure I r (8) . Dualising this with thereal part of (4,0)-form, one concludes that for each N ( I r (8) ) there is a 1-form on X . Moreoverit turns out that N ( I r (8) ) is ∇ (+) -parallel. For this, one again uses hol( ∇ (+) ) ⊆ K , the identity(3.2), dH = 0 and the field equations, where K is Sp (2), × Sp (1), Sp (1) or U (1). As a resultfor every non-vanishing Nijenhuis tensor N ( I r (8) ), there is an associated ∇ (+) -parallel 1-form. Asimilar calculation also reveals that τ I r (8) are also ∇ (+) -parallel. Of course the commutators ofthe associated vector fields, if they are non-vanishing, they are also ∇ (+) -parallel. If one or moresuch forms are non-vanishing, the structure groups reduce. In particular for Sp (2), if there areone or more linearly independent ∇ (+) -parallel 1-forms τ , the structure group reduces to either Sp (1) = SU (2) or { } . The reduction pattern for × Sp (1) is similar. In the Sp (1) case, if thereis parallel 1-form, the structure group reduces to the identity. A similar result holds for the25 (1) case.Step 2Now suppose that hol( ∇ (+) ) ⊆ SU (2). Observe that the action of this SU (2) is different fromthat of Sp (1), which is associated with 5 parallel spinors, on the typical fibre of T X . Inparticular, the spacetime R , × X with hol( ∇ (+) ) ⊆ SU (2) admits 8 parallel spinors. Thesegive rise to four ∇ (+) -parallel forms, e , e , e and e , and Hermitian forms ω I (4) and ω J (4) on X obeying the algebraic relations which have already been stated in the investigation of thereduction of the SU (4)-structure to SU (2). The details of the analysis are similar to those ofthe SU (4) case and so we shall not expand further here. This similarity extends whenever thestructure groups reduce to { } . Eight-dimensional manifolds with a
Spin (7)-structure have been investigated in [39]. It is knownthat any 8-dimensional manifold with
Spin (7)-structure admits a compatible connection withskew-symmetric torsion [40]. So the gravitino KSE can be solved for every 8-dimensional mani-fold which admits a
Spin (7) structure. Moreover, the torsion is completely determined in termsof the metric and fundamental self-dual 4-form φ as H = − ⋆ dφ + ⋆ ( θ ∧ φ ) , (5.41)where θ φ = − ⋆ ( ⋆dφ ∧ φ ) is the Lee 1-form. The torsion 3-form is not always closed.The dilatino KSE requires that τ φ , (5.42)must vanish.Step 1Now suppose that we have a solution of the gravitino KSE only, and so hol( ∇ (+) ) ⊆ Spin (7).Then using, hol( ∇ (+) ) ⊆ Spin (7), dH = 0, the identity (3.2) and the field equations, one findsthat ∇ (+) τ φ = 0 . (5.43)It is clear that if the dilatino KSE is satisfied, ie τ φ = 0, there is no reduction of the Spin (7)structure of X . However if the dilatino KSE is not satisfied, and so τ φ = 0, the holonomyof ∇ (+) reduces to a subgroup of Spin (7). To identify this subgroup, note that
Spin (7) actswith the spinor representation on the typical fibre of
T X , and there is one type of a non-trivialorbit with isotropy group G . Therefore the holonomy of ∇ (+) and so the structure group of X reduces to a subgroup of G , ie hol( ∇ (+) ) ⊆ G .Step 2Now if hol( ∇ (+) ) ⊆ G , the gravitino KSE of X admits an additional ∇ (+) -parallel spinor. Insuch case, the ∇ (+) -parallel forms bilinears on X are an 1-form e , and the fundamental G ϕ and ⋆ ϕ , respectively, such that i ϕ = i ⋆ ϕ = 0, where the subscripted stardenotes the Hodge duality operation in directions orthogonal to e . Since the reduction to G has been mediated by the non-vanishing τ φ e = τ φ .26dapting a local frame as ( e i , e ), the metric and 3-form are written as ds = ( e ) + δ ij e i e j , i, j = 1 , . . . , ,H = 12 H ij e ∧ e i ∧ e j + 13! H ijk e i ∧ e j ∧ e k . (5.44)Applying the results of [21] on the manifold R , × X , where the metric and torsion on X aregiven as above, the gravitino KSE requires the conditions( de ) ij | = 16 ∇ ϕ mn [ i ϕ mnj ] , (5.45)and i H = de , in addition to those stated in section 5.1.3 for the directions orthogonal to e .The dilatino KSE for the (5.44) background implies that ∂ Φ , ˜ de | , τ ϕ , (5.46)must vanish, where ˜ d is the exterior derivative restricted to directions orthogonal to e . If ∂ Φ = 0, the G does not reduce further. However if either ˜ de | = 0 or τ ϕ = 0, the structurereduces to SU (3). If both do not vanish and are linearly independent, then the structure reducesto SU (2).Step 3It remains to investigate the further reduction of SU (3)- and SU (2)-structures. The analysis issimilar to that that we have already described in the SU (4) case in section 5.2.1, so we shall notpursue this further. In this paper we have seen that when H -flux is turned on many of the special holonomy algebrasare potentially deformed by currents that are associated with generalised Nijenhuis forms. Thegeneral analysis from the worldsheet perspective is rather complicated in arbitrary dimension inthe absence of integrability, but as demonstrated in this paper it is tractable in lower dimensions.Fortunately these are also the physically relevant cases. We were also able to analyse theimplications for the worldsheet algebras due to various covariantly constant one-forms whosepresence is justified by imposing conformal invariance, i.e. assuming the stringy equations ofmotion.The general situation is that there are further reductions in the structure group that was initiallytaken to be determined by the presence of special holonomy forms. In the second part of thepaper, adopting a spacetime point of view, we classified all possible reductions of this type usingthe Killing Spinor Equations. Acknowledgements:
GP is partially supported by the EPSRC grant EP/F069774/1 and the STFC rolling grantST/G000/395/1. VS was supported by the DFG (German Science Foundation) and the Univer-sity of Hamburg, as well as, in part, by a NWO VICI grant. VS is also grateful for the supportof Per Sundell and the University of Mons (F.R.S.-FNRS Ulysse Incentive Grant for Mobility inScientific Research). 27
Notation and Conventions (1 ,
1) superspace has coordinates z = ( x ++ , x −− , θ + , θ − ). D + and D − are the usual flat super-space covariant derivatives which obey the relations D = i∂ ++ ; D − = i∂ −− ; { D + , D − } = 0 . (A.1)We use the convention that ∂ ++ x ++ = 1.The action for a (1 , S = Z dz ( g ij + b ij ) D + X i D − X j , (A.2)where dz := d x D + D − . The action (A.2) is invariant under superconformal transformationswhich act independently on the left (+) and right (-) light-cone sectors. In the left sector, thesupercurrent is the energy-momentum tensor T +3 := g ij ∂ ++ X i D + X j − i H ijk D + X ijk , (A.3)where we have introduced the abbreviation D + X ijk := D + X i D + X j D + X k . (A.4)The current is conserved in the sense that D − T +3 = 0 on-shell. Similarly, there is a conservedenergy-momentum tensor T − in the right sector.The action of the (1 ,
0) model is given by S = Z dx ++ dx −− dθ + ( g ij + b ij ) D + X i ∂ −− X j . (A.5)In the left sector we have a supercurrent as in the (1 ,
1) model, (A.3), but in the right sectorthe conserved current is just: T − = g ij ∂ ++ X i ∂ ++ X j . (A.6)Let L be a vector-valued l -form such that the l + 1-form obtained by lowering the vector index(taken to be in the first slot) is covariantly constant with respect to ∇ (+) ; this form will also bedenoted L . (It should be clear from the context which is meant). The actions of the (1 ,
1) andthe (1 ,
0) models are both are invariant under the transformation δ L X i = a L L iL D + X L , (A.7)where the parameter a L has Lorentz weight − l , Grassmann parity ( − l and is chiral, D − a L = 0.The multi-index L denotes l antisymmetrised indices, L := [ l . . . l l ]. We shall use the notation L to denote antisymmetrisation over the l − l , and so on. Analogous L -type symmetries exist in the right sector only in the case of the (1 ,
1) model.28
Factorization of geometries with special holonomy
Here we shall show that if hol( ∇ (+) ) ⊆ × Sp (1) and two commuting complex structures areintegrable, then X locally factorizes as X = X × X ′ , where X and X ′ admit an Sp (1)structure compatible with a connection with skew-symmetric torsion. Moreover under the sameassumption, if ∇ (+) has holonomy either Sp (1) ⊂ × Sp (1) or U (1) ⊂ Sp (1) ⊂ × Sp (1), then X is parallelisable.First let us begin with × Sp (1). This follows from the more general result that if the holonomyof ∇ (+) is U ( n ) × U ( m ) and the associated complex and product structures are integrable, then X n +2 m locally factorizes as X n +2 m = X n × X m , where where X n and X m admit a U ( n )and a U ( m ) structure compatible with a connection with skew-symmetric torsion, respectively.Indeed if ∇ (+) has holonomy U ( n ) × U ( m ), it admits two commuting complex structures I, J , I = J = − IJ = J I . Π = IJ is a product structure and it is integrable provided thatboth I and J are integrable. In such a case, there is an atlas on X n +2 m such that I = ( iδ αβ , iδ µν , − iδ ¯ α ¯ β , − iδ ¯ µ ¯ ν ) ,J = ( iδ αβ , − iδ µν , − iδ ¯ α ¯ β , iδ ¯ µ ¯ ν ) , (B.1)where ( z α , w µ ), α = 1 , . . . , n , µ = 1 , . . . , m , are holomorphic coordinates and the transitionfunctions are holomorphic respecting the splitting. The integrability of the complex structuresimplies that the non-vanishing components of H are H αβ ¯ γ , H µ ¯ να , H α ¯ βµ , H µν ¯ ρ , (B.2)and their complex conjugates. Since the metric is hermitian with respect to both complexstructures, the non-vanishing components of the metric are g = ( g α ¯ β , g µ ¯ ν ) . (B.3)So far the components of the torsion H and the metric depend on all coordinates.Since I and J are ∇ (+) -covariantly constant, H is determined in terms of both the complexstructures I and J leading to the condition H = − i I dω I = − i J dω J . (B.4)Evaluating H µ ¯ να using both the I and J complex structures, one finds that H µ ¯ να = − ∂ α g µ ¯ ν and H µ ¯ να = 3 ∂ α g µ ¯ ν , respectively. Therefore consistency requires that ∂ α g µ ¯ ν = 0 , H µ ¯ να = 0 , (B.5)and similarly ∂ µ g α ¯ β = 0 , H α ¯ βµ = 0 . (B.6)As a result, X n +2 m is metrically locally a product, X n +2 m = X n × X m . X n is a hermitianmanifold with complex structure I = ( iδ αβ , − iδ ¯ α ¯ β ) and metric g = ( g α ¯ β ) and so with torsion H = ( H αβ ¯ γ , H ¯ α ¯ βγ ), where all components of g and H depend only on the coordinates ( z α , z ¯ α ).Similarly, X m is a hermitian manifold with complex structure I = ( iδ µν , − iδ ¯ µ ¯ ν ) and metric g = ( g µ ¯ ν ) and torsion H = ( H µν ¯ ρ , H ¯ µ ¯ νρ ), where all components of g and H depend only onthe coordinates ( w µ , w ¯ µ ). 29 consequence of the result above is that if X has holonomy × Sp (1) ⊂ × U (2), then X is the product of two 4-dimensional HKT manifolds. 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