On-line version ISSN 1669-9637
Rev. Unión Mat. Argent. vol.49 no.2 Bahía Blanca July/Dec. 2008
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 hyper-Kähler 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 hyper-Hermitian 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 skew-symmetric, for any . The hyper-Hermitian manifold is said to be hyperkähler with torsion (HKT for short)  if there exists a hyper-Hermitian 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 hyper-Kähler geometry. In fact, when the -form associated to an HKT structure vanishes, then the connection coincides with the Levi-Civita connection and the metric is hyper-Kähler.
An HKT structure is called strong or weak depending on whether the -form is closed or not.
In  it was proved that the HKT condition is equivalent to
where are the associated Kähler forms
Given a Hermitian manifold , there exists a unique Hermitian connection such that
The study of hyper-Hermitian 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  that the geometry of the moduli space of a class of black holes in five dimensions is hyper-Kähler with torsion (see also ).
Many examples of HKT manifolds have been obtained. A twistor construction of HKT manifolds was proposed in  and HKT reduction has been studied in  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  and independently by Joyce in , which was generalized in  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 . Using results of , it was shown in  that carries inhomogeneous weak HKT structures. Also, inhomogeneous examples of compact HKT manifolds which are not locally conformal hyper-Kähler can be obtained by considering the total space of a hyperholomorphic bundle over a compact HKT manifold .
Some geometrical and topological properties have been investigated. Differential geometric properties of HKT manifolds and their twistor spaces have been studied in  and it was proved in  that, in analogy to the hyper-Kähler 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 . A version of Hodge theory for HKT manifolds has been given in  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  balanced HKT metrics were studied, showing that the HKT metrics are precisely the quaternionic Calabi-Yau metrics defined in terms of the quaternionic Monge-Ampère equation. Moreover, by  a balanced HKT manifold has Obata connection with holonomy in
We recall first the properties of the de Rham algebra of a Kähler manifold (see, for instance, ). 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:
Using the Kähler form , it is possible to define the following linear operators:
where is the Hodge-star operator. When is Kähler, the operators
satisfy the Kodaira relations (see ). For instance, one has
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 ) Let be a compact Kähler manifold with Kähler form . Then, for any , the map
Given an HKT manifold , it is shown in  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 hyper-Hermitian 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
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.
An HKT structure on a Lie group is called left-invariant when left translations , , are isometries and holomorphic maps with respect to for any . In this case, it has been shown in  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  (see also ), where is a compact semisimple Lie group. In this case, the restriction of the HKT metric to is the opposite of the Killing-Cartan 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 .
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 skew-symmetric 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. □
In  strong HKT structures have been constructed on non-semisimple 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 ) when for all (resp. ). Observe that in this case (11) is automatically satisfied for any hyper-Hermitian metric , that is, given an abelian hypercomplex structure, any hyper-Hermitian metric is HKT. Moreover, it was shown in [9, Proposition 2.1] that left-invariant 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 left-invariant HKT structure arises from an abelian hypercomplex structure. We proved in  that this theorem still holds for -step nilpotent Lie groups admitting lattices, for arbitrary (see Theorem 4.2).
We point out that the left-invariant complex structure on gives rise to a decomposition
where are the eigenspaces of the induced complex structure on . It turns out that is abelian if and only if is an abelian subalgebra of .
A nilmanifold (see) is a quotient of a simply connected nilpotent Lie group by a lattice (a discrete co-compact subgroup). It is well known that
- admits lattices if and only if has a rational form.
Moreover, there is a one-to-one correspondence:
Let be a nilmanifold and assume that is equipped with a left-invariant complex structure . Then induces a complex structure on . A complex structure on is called abelian if it is induced from a left-invariant abelian complex structure on .
The first example of symplectic non-Kähler manifold was described by Thurston : 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 ):
It follows that:
- covers for any .
The nilmanifolds have fundamental group isomorphic to , in particular, they are not homeomorphic. These are examples of symplectic non-Kähler manifolds. In the 80's, many authors (Abbena, Cordero, Fernández, Gray, de León, among others) obtained families of symplectic non-Kähler manifolds as generalizations of the previous example. Later, in 1988, the following remarkable theorem was proved by Benson-Gordon (see also , 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:
- 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.
More precisely, Benson-Gordon show that if is non-abelian nilpotent, then the map
Theorem 4.2 (). Let be a -dimensional nilmanifold endowed with an HKT structure induced by a left-invariant HKT structure on . Then is abelian.
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 :
- Given a compact manifold with a hypercomplex structure, is it always possible to find a compatible HKT metric?
A negative answer was given in  by exhibiting -step nilmanifolds with non-abelian hypercomplex structures. In view of Theorem 4.2, any non-abelian 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 ).
This class of Lie algebras has first been considered in .
We observe that:
- is a nilpotent Lie algebra if and only if is nilpotent as an associative algebra.
Let be the endomorphism of defined by:
It has been shown in  that is a complex structure on . If, moreover, is a complex associative algebra, we can define another complex structure on by:
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:
- The hypercomplex -step nilmanifold Aff does not admit a compatible HKT metric.
Remark 4.2. We point out that for the Lie algebra is not two-step solvable, hence it does not admit abelian hypercomplex structures (see ). Therefore, Theorem implies that any left-invariant hypercomplex structure on Aff, , induces on the nilmanifold a hypercomplex structure admitting no compatible HKT metric.
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Universidad Nacional de Córdoba,
Córdoba 5000, Argentina
Recibido: 26 de noviembre de 2008
Aceptado: 26 de noviembre de 2008