M. L. Barberis
Abstract. Manifolds with special geometric structures play a prominent role in some branches of theoretical physics, such as string theory and supergravity. For instance, it is well known that supersymmetry requires target spaces to have certain special geometric properties. In many cases these requirements can be interpreted as restrictions on the holonomy group of the target space Riemannian metric. However, in some cases, they cannot be expressed in terms of the Riemannian holonomy group alone and give rise to new geometries previously unknown to mathematicians. An example of this situation is provided by hyperKähler with torsion (or HKT) metrics, a particular class of metrics which possess a compatible connection with torsion whose holonomy lies in Sp(n).
A survey on recent results on HKT geometry is presented.
A hyperHermitian structure on a dimensional manifold is given by a hypercomplex structure , (a triple of complex structures satisfying the imaginary quaternion relations) and a Riemannian metric with respect to which is skewsymmetric, for any . The hyperHermitian manifold is said to be hyperkähler with torsion (HKT for short) [19] if there exists a hyperHermitian connection whose torsion tensor is a form, that is,

where is the torsion of . It follows from the definition that the holonomy of lies in . HKT geometry is a generalization of hyperKähler geometry. In fact, when the form associated to an HKT structure vanishes, then the connection coincides with the LeviCivita connection and the metric is hyperKähler.
An HKT structure is called strong or weak depending on whether the form is closed or not.
In [16] it was proved that the HKT condition is equivalent to
]]> (2) 
where are the associated Kähler forms

Also, a holomorphic characterization has been given in [16], where the authors proved that (2) is equivalent to

More recently, in [24], it has been shown that if is almost hyperHermitian, then condition (2) implies the integrability of
Given a Hermitian manifold , there exists a unique Hermitian connection such that
The study of hyperHermitian connections satisfying (1) is motivated by the fact that these structures appear in some branches of theoretical physics, such as string theory, in the context of certain supersymmetric sigma models [11, 19, 20, 28]. These connections are also present in supergravity theories. For instance, it has been shown in [13] that the geometry of the moduli space of a class of black holes in five dimensions is hyperKähler with torsion (see also [27]).
Many examples of HKT manifolds have been obtained. A twistor construction of HKT manifolds was proposed in [19] and HKT reduction has been studied in [15] in order to construct new examples. A large family of strong HKT manifolds is given by compact Lie groups with the hypercomplex structure constructed in [31] and independently by Joyce in [22], which was generalized in [26] to the case of homogeneous spaces. On the other hand, there are partial results concerning HKT structures on solvable Lie groups, where weak examples abound [4, 9]. Strong HKT structures on Lie groups with compact Levi factor have been obtained in [5]. Using results of [29], it was shown in [16] that carries inhomogeneous weak HKT structures. Also, inhomogeneous examples of compact HKT manifolds which are not locally conformal hyperKähler can be obtained by considering the total space of a hyperholomorphic bundle over a compact HKT manifold [34].
]]> Some geometrical and topological properties have been investigated. Differential geometric properties of HKT manifolds and their twistor spaces have been studied in [21] and it was proved in [1] that, in analogy to the hyperKähler case, locally any HKT metric admits an HKT potential. A simple characterization of HKT geometry in terms of the intrinsic torsion of the structure was obtained in [24]. A version of Hodge theory for HKT manifolds has been given in [33] by exploiting a remarkable analogy between the de Rham complex of a Kähler manifold and the Dolbeault complex of an HKT manifold. More recently, in [35] balanced HKT metrics were studied, showing that the HKT metrics are precisely the quaternionic CalabiYau metrics defined in terms of the quaternionic MongeAmpère equation. Moreover, by [35] a balanced HKT manifold has Obata connection with holonomy in2. Hodge theory on HKT manifolds
We review in this section some fundamental facts from the theory developed by Verbitsky in [33] that will be relevant to explain the main result obtained in [4] (see §4).
We recall first the properties of the de Rham algebra of a Kähler manifold (see, for instance, [17]). Let be a Hermitian manifold, that is, is a complex structure on and is a Riemannian metric such that for all vector fields on .
The Kähler form is defined as in (3) and is Kähler if and only if is closed. acts on differential forms as
with . Let be the following differential operator acting on forms:
so that

where is the Hodgestar operator. When is Kähler, the operators

satisfy the Kodaira relations (see [17]). For instance, one has
and, moreover,
that is, when is Kähler, induce an action of on the complex cohomology of .
There are some cohomological restrictions imposed by the existence of a Kähler metric on a compact manifold. One necessary condition is that the odd Betti numbers must be even. The following classical result gives another cohomological condition satisfied by compact Kähler manifolds.
]]> Hard Lefschetz Theorem. (See [17]) Let be a compact Kähler manifold with Kähler form . Then, for any , the mapGiven an HKT manifold , it is shown in [33] that the Dolbeault differential graded algebra is an analogue of the de Rham algebra of a Kähler manifold. The roles of the de Rham differential and of the Kähler form are played by and by the form defined in (7) below, respectively. One can associate with three operators as in (5), thereby obtaining an action of on .
Let be a hyperHermitian structure on a dimensional manifold and consider the following form with respect to :

Using it is possible to construct three linear operators as in (5). We denote by the forms of type with respect to . Let

be the Dolbeault operator with respect to the complex structure and

It was shown in [33] that on an HKT manifold, plays the role of on a Kähler manifold. In fact, it follows from [33, Corollary 7.2] that on an HKT manifold, the operatots
]]> (10) 
satisfy the same identities which hold for the operators (6).
The next result, which is a particular case of [33, Theorem 10.2] and is one of the main steps in the proof of Theorem 4.2, is an analogue of the Hard Lefschetz Theorem for the Dolbeault cohomology of HKT manifolds.
Theorem 2.1 ([33]). Let be a compact dimensional HKT manifold with form as in (7) and assume that the canonical bundle is holomorphically trivial. Then, for any :
3. HKT structures on Lie groups
An HKT structure on a Lie group is called leftinvariant when left translations , , are isometries and holomorphic maps with respect to for any . In this case, it has been shown in [9] that the HKT condition is equivalent to: for any , the Lie algebra of .
A large family of HKT manifolds is provided by with the hypercomplex structure obtained in [31] (see also [22]), where is a compact semisimple Lie group. In this case, the restriction of the HKT metric to is the opposite of the KillingCartan form and the Bismut connection is the canonical affine connection on defined by

Conversely, if (12) is the Bismut connection of some left invariant KT metric, then is isomorphic to a direct product of an abelian Lie group by a compact semisimple Lie group (Corollary 3.1). This fact is a consequence of a classical result due to Milnor [25].
]]> Lemma 3.1. Let be a connected Lie group with a left invariant metric . Then 
is a form if and only if is isomorphic to , where is a compact connected Lie group.
Proof. We observe that is a form if and only if is skewsymmetric for any . It follows from [25, Lemmas 7.2 and 7.5] that this occurs if and only if is as in the statement. □
Corollary 3.1. Let be a connected Lie group and the canonical connection (12) on . If is the KT connection associated to some left invariant Hermitian metric on , then is isomorphic to , where is a compact connected Lie group.
Proof. We observe that the torsion of is . Therefore, if is the Bismut connection of , we must have that is a form and the corollary follows from Lemma 3.1. □
As a consequence of the above corollary, one has that if the canonical connection defined in (12) is the Bismut connection of an HKT structure on , then is as in Corollary 3.1.
In [5] strong HKT structures have been constructed on nonsemisimple Lie groups starting with a compact Lie group acting on by quaternionic linear maps which are isometries of the Euclidean metric.
A left invariant complex (resp. hypercomplex) structure on is called abelian (see [2]) when for all (resp. ). Observe that in this case (11) is automatically satisfied for any hyperHermitian metric , that is, given an abelian hypercomplex structure, any hyperHermitian metric is HKT. Moreover, it was shown in [9, Proposition 2.1] that leftinvariant HKT structures arising from abelian hypercomplex structures are always weak.
It was shown in [9, Theorem 3.1] that for step nilpotent Lie groups every leftinvariant HKT structure arises from an abelian hypercomplex structure. We proved in [4] that this theorem still holds for step nilpotent Lie groups admitting lattices, for arbitrary (see Theorem 4.2).
]]> We point out that the leftinvariant complex structure on gives rise to a decompositionwhere are the eigenspaces of the induced complex structure on . It turns out that is abelian if and only if is an abelian subalgebra of .
4. HKT structures on nilmanifolds
4.1. Generalities on nilmanifolds.
A nilmanifold (see[23]) is a quotient of a simply connected nilpotent Lie group by a lattice (a discrete cocompact subgroup). It is well known that
 admits lattices if and only if has a rational form.
Moreover, there is a onetoone correspondence:
The first example of symplectic nonKähler manifold was described by Thurston [32]: it is the nilmanifold , where
is the dimensional Heisenberg group and is the subgroup of matrices in with integer entries.
For each one can define a lattice in as follows (compare with [14]):
It follows that:
 covers for any .
 . ]]>
 The de Rham cohomology of can be identified with the Lie algebra cohomology of due to a result of Nomizu.
 It is proved that if is nilpotent, the Hard Lefschetz Theorem implies that is abelian.
 Given a compact manifold with a hypercomplex structure, is it always possible to find a compatible HKT metric? ]]>
 is a nilpotent Lie algebra if and only if is nilpotent as an associative algebra.
 The hypercomplex step nilmanifold Aff does not admit a compatible HKT metric.
The nilmanifolds have fundamental group isomorphic to , in particular, they are not homeomorphic. These are examples of symplectic nonKähler manifolds. In the 80's, many authors (Abbena, Cordero, Fernández, Gray, de León, among others) obtained families of symplectic nonKähler manifolds as generalizations of the previous example. Later, in 1988, the following remarkable theorem was proved by BensonGordon (see also [18], where the author showed that a minimal model of a nilmanifold is formal if and only if it is a torus):
Theorem 4.1 ([6, Theorem A]). If is a Kähler nilmanifold, then is abelian and is diffeomorphic to a torus.
The main ingredients in the proof of the above theorem are:
More precisely, BensonGordon show that if is nonabelian nilpotent, then the map
For the case of HKT nilmanifolds, we proved in [4] the following analogue of Theorem 4.1:
]]> Theorem 4.2 ([4]). Let be a dimensional nilmanifold endowed with an HKT structure induced by a leftinvariant HKT structure on . Then is abelian.Sketch of proof. The aim is to show that is abelian (equivalently, is abelian). We observe that the canonical bundle of is trivial [8], therefore Theorem 2.1 applies.
Consider the following commutative diagram:
where the vertical arrows are the natural identifications. The Lie algebra is nilpotent. If we assume that is not abelian, one can apply the same argument in [6, Lemma 2.11] to the bottom row of the previous diagram to obtain that
The following question was posed in [16]:
A negative answer was given in [10] by exhibiting step nilmanifolds with nonabelian hypercomplex structures. In view of Theorem 4.2, any nonabelian hypercomplex structure on a step nilmanifold admits no compatible HKT metric. Therefore, Theorem 4.2 provides a useful tool for obtaining many examples where the answer to the above question is negative. To illustrate this situation, we exhibit next a family of hypercomplex step nilmanifolds, for arbitrary , admitting no compatible HKT metric (see [4]).
4.2. A family of examples. Let be a finite dimensional associative algebra and let be the Lie algebra with Lie bracket:
This class of Lie algebras has first been considered in [3].
We observe that:
Let be the endomorphism of defined by:

Remark 4.1. is abelian if and only if is conmutative.
Let be the algebra of strictly upper triangular matrices with complex entries and consider the simply connected Lie group Aff with Lie algebra , which is step nilpotent. The structure constants of with respect to the standard basis are integers, hence Aff admits a lattice and we obtain:
Remark 4.2. We point out that for the Lie algebra is not twostep solvable, hence it does not admit abelian hypercomplex structures (see [30]). Therefore, Theorem implies that any leftinvariant hypercomplex structure on Aff, , induces on the nilmanifold a hypercomplex structure admitting no compatible HKT metric.
[1] B. Banos, A. Swann, Potentials for hyperKähler metrics with torsion, Class. Quant.Grav. 21 (2004) 31273136. [ Links ]
[2] M.L. Barberis, I. Dotti, R. Miatello, On certain locally homogeneous Clifford manifolds, Ann. Glob. Anal. Geom. 13 (1995), 289301. [ Links ]
[3] M.L. Barberis, I. Dotti, Abelian complex structures on solvable Lie algebras, J. Lie Theory 14 (1) (2004), 2534. [ Links ]
[4] M.L. Barberis, I. Dotti, M. Verbitsky, Canonical bundles of complex nilmanifolds, with applications to hypercomplex geometry, arXiv:math.DG/0712.3863, to appear in Math. Res. Lett. [ Links ]
[5] M.L. Barberis, A. Fino, New strong HKT manifolds arising from quaternionic representations, arXiv:0805.2335. [ Links ]
[6] C. Benson, C.S. Gordon, Kähler and symplectic structures on nilmanifolds, Topology 27(4) (1988) 513518. [ Links ]
[7] J. M. Bismut, Local index theorem for nonKähler manifolds, Math. Ann. 284 (1989), 681699. [ Links ]
[8] G.R. Cavalcanti, M. Gualtieri, Generalized complex structures on nilmanifolds, arXiv:math/0404451, J. Symplectic Geom. 2 (3) (2004), 393410. [ Links ]
[9] I. Dotti, A. Fino, HyperKähler with torsion structures invariant by nilpotent Lie groups, Class. Quantum Grav. 19 (2002), 112. [ Links ]
[10] A. Fino, G. Grantcharov, Properties of manifolds with skewsymmetric torsion and special holonomy, Adv. Math. 189 (2004), 439450. [ Links ]
[11] S. J. Gates, C.M. Hull, M. Roček, Twisted multiplets and new supersymmetric nonlinear models, Nucl. Phys. B 248 (1984), 157186. [ Links ]
[12] P. Gauduchon, K.P. Tod, HyperHermitian metrics with symmetry, J. Geom. Phys. 25 (1998), 291304. [ Links ]
[13] G. W. Gibbons, G. Papadopoulos, K. Stelle, HKT and OKT geometries on soliton blach hole moduli space, Nucl. Phys. B 508 (1997), 623. [ Links ]
[14] C. S. Gordon and E. N. Wilson, The spectrum of the Laplacian on Riemannian Heisenberg manifolds, Michigan Math. J. 33 (2) (1986), 253271. [ Links ]
[15] G. Grantcharov, G. Papadopoulos,, Y. S. Poon, Reduction of HKTStructures, J. Math. Phys. 43 (2002), 37663782. [ Links ]
[16] G. Grantcharov, Y. S. Poon, Geometry of hyperKähler connection with torsion, Comm. Math. Phys. 213 (2000), 1937. [ Links ]
[17] P. Griffiths, J. Harris, Principles of algebraic geometry, WileyInterscience, New York, 1978. [ Links ]
[18] K. Hasegawa, Minimal models of nilmanifolds, Proc. Am. Math. Soc. 106 (1989), 6571. [ Links ]
[19] P.S. Howe, G. Papadopoulos, Twistor spaces for HKT manifolds, Phys. Lett. B 379 (1996), 8186. [ Links ]
[20] P.S. Howe, G. Papadopoulos, Finiteness and anomalies in suspersymmetric sigma models for HKT manifolds, Nucl. Phys. B 381 (1992), 360372.
[21] S. Ivanov, I. Minchev, Quaternionic Kähler and hyperKähler manifolds with torsion and twistor spaces, J. Reine Angew. Math. 567 (2004), 215233. [ Links ]
[22] D. Joyce, Compact hypercomplex and quaternionic manifolds, J. Diff. Geom. 35 (1992), 743761. [ Links ]
[23] A. I. Mal'čev, On a class of homogeneous spaces, AMS Translation No. 39 (1951). [ Links ]
[24] F. Martín Cabrera, A. Swann, The intrinsic torsion of almost quaternionHermitian manifolds, arXiv:math.DG/0707.0939, to appear in Annales de l'Institute Fourier. [ Links ]
[25] J. Milnor, Curvatures of left invariant metrics on Lie groups, Adv. Math. 21 (3) (1976), 293329. [ Links ]
[26] A. Opfermann, G. Papadopoulos, Homogeneous HKT and QKT manifolds, arXiv:mathph9807026. [ Links ]
[27] G. Papadopoulos, A. Teschendorff, Multi angle five brane intersections, Phys. Lett. B 443 (1998), 159. [ Links ]
[28] G. Papadopoulos, KT and HKT geometries in strings and in black hole moduli spaces, Proceedings of the Bonn workshop on "Special Geometric Structures in String Theory", 811 September 2001, Eds.: D.V. Alekseevsky, V. Cortés, C. Devchand, A. Van Proeyen, arXiv:hepth/0201111. [ Links ]
[29] H. Pedersen, Y.S. Poon, Inhomogeneous hypercomplex structures on homogeneous manifolds, J. Reine Angew. Math. 516 (1999), 159181. [ Links ]
[30] P. Petravchuk, Lie algebras decomposable as a sum of an abelian and a nilpotent subalgebra, Ukr. Math. J. 40(3) (1988), 385388. [ Links ]
[31] Ph. Spindel, A. Sevrin, W. Troost, A. Van Proeyen, Extended supersymmetric models on group manifolds, Nucl. Phys. B 308 (1988), 662698. [ Links ]
[32] W.P. Thurston, Some simple examples of symplectic manifolds, Proc. Am. Math. Soc. 55 (1976), 476478. [ Links ]
[33] M. Verbitsky, Hyperkähler manifolds with torsion, supersymmetry and Hodge theory, Asian J. Math. 6 (2002), 679712 . [ Links ]
[34] M. Verbitsky, Hyperkähler manifolds with torsion obtained from hyperholomorphic bundles, Math. Res. Lett. 10 (2003), 501513. [ Links ]
[35] M. Verbitsky, Balanced HKT metrics and strong HKT metrics on hypercomplex manifolds preprint math.DG/0808.3218. [ Links ]
Laura Barberis
FaMAFCIEM,
Universidad Nacional de Córdoba,
Córdoba 5000, Argentina
barberis@mate.uncor.edu
Recibido: 26 de noviembre de 2008 ]]> Aceptado: 26 de noviembre de 2008
]]>