The Hermitian Laplace Operator on Nearly Kähler Manifolds

The moduli space ${\mathcal {NK}}The moduli space NK{\mathcal {NK}} of infinitesimal deformations of a nearly K?hler structure on a compact 6-dimensional manifold is described by a certain eigenspace of the Laplace operator acting on co-closed primitive (1, 1) forms (cf. Moroianu et al. in Pacific J Math 235:57–72, 2008). Using the Hermitian Laplace operator and some representation theory, we compute the space NK{\mathcal {NK}} on all 6-dimensional homogeneous nearly K?hler manifolds. It turns out that the nearly K?hler structure is rigid except for the flag manifold F(1, 2) = SU₃/T ², which carries an 8-dimensional moduli space of infinitesimal nearly K?hler deformations, modeled on the Lie algebra \mathfraksu₃{\mathfrak{su}_3} of the isometry group.     

Yang-Mills Flows on Nearly Kähler Manifolds and G 2-Instantons

We consider Lie(G)-valued G-invariant connections on bundles over spaces ${G/H,\, \mathbb{R}\times G/H\, {\rm and}\, \mathbb{R}^2\times G/H}We give a geometric construction of the ${\mathcal{W}_{1+\infty}}We consider Lie(G)-valued G-invariant connections on bundles over spaces G/H, \mathbbR×G/H and \mathbbR²×G/H{G/H,\, \mathbb{R}\times G/H\, {\rm and}\, \mathbb{R}^2\times G/H}, where G/H is a compact nearly K?hler six-dimensional homogeneous space, and the manifolds \mathbbR×G/H{\mathbb{R}\times G/H} and \mathbbR²×G/H{\mathbb{R}^2\times G/H} carry G ₂- and Spin(7)-structures, respectively. By making a G-invariant ansatz, Yang-Mills theory with torsion on \mathbbR×G/H{\mathbb{R}\times G/H} is reduced to Newtonian mechanics of a particle moving in a plane with a quartic potential. For particular values of the torsion, we find explicit particle trajectories, which obey first-order gradient or hamiltonian flow equations. In two cases, these solutions correspond to anti-self-dual instantons associated with one of two G ₂-structures on \mathbbR×G/H{\mathbb{R}\times G/H}. It is shown that both G ₂-instanton equations can be obtained from a single Spin(7)-instanton equation on \mathbbR²×G/H{\mathbb{R}^2\times G/H}.     

Special Kähler Manifolds

We give an intrinsic definition of the special geometry which arises in global N= 2 supersymmetry in four dimensions. The base of an algebraic integrable system exhibits this geometry, and with an integrality hypothesis any special K?hler manifold is so related to an integrable system. The cotangent bundle of a special K?hler manifold carries a hyperk?hler metric. We also define special geometry in supergravity in terms of the special geometry in global supersymmetry. Received: 5 December 1997 / Accepted: 16 November 1998     

Six-Dimensional Nearly Kähler Manifolds of Cohomogeneity One (II)

Let M be a six dimensional manifold, endowed with a cohomogeneity one action of G = SU₂ × SU₂, and \({M_{\rm reg} \subset M}\) its subset of regular points. We show that M _reg admits a smooth, 2-parameter family of G-invariant, non-isometric strict nearly Kähler structures and that a 1-parameter subfamily of such structures smoothly extends over a singular orbit of type S ³. This determines a new class of examples of nearly Kähler structures on T S ³.     

Conification of Kähler and Hyper-Kähler Manifolds

Given a Kähler manifold M endowed with a Hamiltonian Killing vector field Z, we construct a conical Kähler manifold ${\hat{M}}$ such that M is recovered as a Kähler quotient of ${\hat{M}}$ . Similarly, given a hyper-Kähler manifold (M, g, J ₁, J ₂, J ₃) endowed with a Killing vector field Z, Hamiltonian with respect to the Kähler form of J ₁ and satisfying ${\mathcal{L}_ZJ_2 = -2J_3}$ , we construct a hyper-Kähler cone ${\hat{M}}$ such that M is a certain hyper-Kähler quotient of ${\hat{M}}$ . In this way, we recover a theorem by Haydys. Our work is motivated by the problem of relating the supergravity c-map to the rigid c-map. We show that any hyper-Kähler manifold in the image of the c-map admits a Killing vector field with the above properties. Therefore, it gives rise to a hyper-Kähler cone, which in turn defines a quaternionic Kähler manifold. Our results for the signature of the metric and the sign of the scalar curvature are consistent with what we know about the supergravity c-map.     

Deformations of Nearly Kähler Instantons

We characterize the absolutely continuous spectrum of the one-dimensional Schrödinger operators \({h = -\Delta + v}\) acting on \({\ell^2(\mathbb{Z}_+)}\) in terms of the limiting behaviour of the Landauer–Büttiker and Thouless conductances of the associated finite samples. The finite sample is defined by restricting h to a finite interval \({[1, L] \cap \mathbb{Z}_+}\) and the conductance refers to the charge current across the sample in the open quantum system obtained by attaching independent electronic reservoirs to the sample ends. Our main result is that the conductances associated to an energy interval \({I}\) are non-vanishing in the limit \({L \to \infty}\) iff \({{\rm sp}_{\rm ac}(h) \cap I \neq \emptyset}\). We also discuss the relationship between this result and the Schrödinger Conjecture (Avila, J Am Math Soc 28:579–616, 2015; Bruneau et al., Commun Math Phys 319:501–513, 2013).     

On Certain Kähler Quotients of Quaternionic Kähler Manifolds

We prove that, given a certain isometric action of a two-dimensional Abelian group A on a quaternionic Kähler manifold M which preserves a submanifold N ? M, the quotient M′ = N/A has a natural Kähler structure. We verify that the assumptions on the group action and on the submanifold N ? M are satisfied for a large class of examples obtained from the supergravity c-map. In particular, we find that all quaternionic Kähler manifolds M in the image of the c-map admit an integrable complex structure compatible with the quaternionic structure, such that N ? M is a complex submanifold. Finally, we discuss how the existence of the Kähler structure on M′ is required by the consistency of spontaneous ${\mathcal{N} = 2}$ to ${\mathcal{N} = 1}$ supersymmetry breaking.     

Flows on Quaternionic-Kähler and Very Special Real Manifolds

BPS solutions of 5-dimensional supergravity correspond to certain gradient flows on the product M×N of a quaternionic-Kähler manifold M of negative scalar curvature and a very special real manifold N of dimension n0. Such gradient flows are generated by the ``energy function' f=P<sup>2</sup>, where P is a (bundle-valued) moment map associated to n+1 Killing vector fields on M. We calculate the Hessian of f at critical points and derive some properties of its spectrum for general quaternionic-Kähler manifolds. For the homogeneous quaternionic-Kähler manifolds we prove more specific results depending on the structure of the isotropy group. For example, we show that there always exists a Killing vector field vanishing at a point pM such that the Hessian of f at p has split signature. This generalizes results obtained recently for the complex hyperbolic plane (universal hypermultiplet) in the context of 5-dimensional supergravity. For symmetric quaternionic-Kähler manifolds we show the existence of non-degenerate local extrema of f, for appropriate Killing vector fields. On the other hand, for the non-symmetric homogeneous quaternionic-Kähler manifolds we find degenerate local minima. This work was supported by the priority programme ``String Theory'of the Deutsche Forschungsgemeinschaft.     

Moment Maps,Scalar Curvature and Quantization of Kähler Manifolds

Building on Donaldsons work on constant scalar curvature metrics, we study the space of regular Kähler metrics E, i.e. those for which deformation quantization has been defined by Cahen, Gutt and Rawnsley. After giving, in Sects. 2 and 3 a review of Donaldsons moment map approach, we study the essential uniqueness of balanced basis (i.e. of coherent states) in a more general setting (Theorem 2.5). We then study the space E in Sect.4 and we show in Sect.5 how all the tools needed can be defined also in the case of non-compact manifolds.     

Currents on locally conformally Kähler manifolds

We characterize the existence of a locally conformally Kähler metric on a compact complex manifold in terms of currents, adapting the celebrated result of Harvey and Lawson for Kähler metrics.     

Quaternionic Kähler metrics associated with special Kähler manifolds

We give an explicit formula for the quaternionic Kähler metrics obtained by the HK/QK correspondence. As an application, we give a new proof of the fact that the Ferrara–Sabharwal metric as well as its one-loop deformation is quaternionic Kähler. A similar explicit formula is given for the analogous (K/K) correspondence between Kähler manifolds endowed with a Hamiltonian Killing vector field. As an example, we apply this formula in the case of an arbitrary conical Kähler manifold.     

Note on Dirac–Kähler massless fields

We obtain the canonical and symmetrical Belinfante energy-momentum tensors of Dirac–Kähler’s fields. It is shown that the traces of the energy-momentum tensors are not equal to zero. We find the canonical and Belinfante dilatation currents which are not conserved, but a new conserved dilatation current is obtained. It is pointed out that the conformal symmetry is broken. The canonical quantization is performed and the propagator of the massless fields in the first-order formalism is found.     

Cauchy–Schwarz-type inequalities on Kähler manifolds

In this note we investigate Cauchy–Schwarz-type inequalities for cohomology elements on compact Kähler manifolds, which can be viewed as generalizations of a classical case. We obtain, as a corollary, some Chern number inequalities when the Hodge numbers of Kähler manifolds satisfy certain restrictions. The same argument can also be applied to compact quaternion-Kähler manifolds with positive scalar curvature to obtain a similar result.     

Generalized Kähler Geometry

Generalized Kähler geometry is the natural analogue of Kähler geometry, in the context of generalized complex geometry. Just as we may require a complex structure to be compatible with a Riemannian metric in a way which gives rise to a symplectic form, we may require a generalized complex structure to be compatible with a metric so that it defines a second generalized complex structure. We prove that generalized Kähler geometry is equivalent to the bi-Hermitian geometry on the target of a 2-dimensional sigma model with (2, 2) supersymmetry. We also prove the existence of natural holomorphic Courant algebroids for each of the underlying complex structures, and that these split into a sum of transverse holomorphic Dirac structures. Finally, we explore the analogy between pre-quantum line bundles and gerbes in the context of generalized Kähler geometry.     

Pseudo-Hyperkähler Geometry and Generalized Kähler Geometry

We discuss the conditions for additional supersymmetry and twisted super-symmetry in N = (2, 2) supersymmetric nonlinear sigma models described by one left and one right semi-chiral superfield and carrying a pair of non-commuting complex structures. Focus is on linear non-manifest transformations of these fields that have an algebra that closes off-shell. We find that additional linear supersymmetry has no interesting solution, whereas additional linear twisted supersymmetry has solutions with interesting geometrical properties. We solve the conditions for invariance of the action and show that these solutions correspond to a bi-hermitian metric of signature (2, 2) and a pseudo-hyperkähler geometry of the target space.     

Kähler spinning particles

We construct the U(N) spinning particle theories, which describe particles moving on Kähler spaces. These particles have the same relation to the N = 2 string as usual spinning particles have to the NSR string. We find the restrictions on the target space of the theories coming from supersymmetry and from global anomalies. Finally, we show that the partition functions of the theories agree with what is expected from their spectra, unlike that of the N = 2 string in which there is an anomalous dependence on the proper time.     

Instantons, Monopoles and Toric HyperKähler Manifolds

In this paper, the metric on the moduli space of the k=1 SU(n) periodic instanton – or caloron – with arbitrary gauge holonomy at spatial infinity is explicitly constructed. The metric is toric hyperK?hler and of the form conjectured by Lee and Yi. The torus coordinates describe the residual U(1) ^{n −1} gauge invariance and the temporal position of the caloron and can also be viewed as the phases of n monopoles that constitute the caloron. The (1,1,...,1) monopole is obtained as a limit of the caloron. The calculation is performed on the space of Nahm data, which is justified by proving the isometric property of the Nahm construction for the cases considered. An alternative construction using the hyperK?hler quotient is also presented. The effect of massless monopoles is briefly discussed. Received: 20 November 1998 / Accepted: 11 October 1999     

Superconformal Symmetry and HyperKähler Manifolds with Torsion

The geometry arising from Michelson & Strominger's study of =4B supersymmetric quantum mechanics with superconformal D(2, 1; )-symmetry is a hyperKähler manifold with torsion (HKT) together with a special homothety. It is shown that different parameters are related via changes in potentials for the HKT target spaces. For 0, –1, we describe how each such HKT manifold M ^4m is derived from a space N ^4m–4 which is quaternionic Kähler with torsion and carries an Abelian instanton.