Mass in Kähler Geometry

We prove a simple, explicit formula for the mass of any asymptotically locally Euclidean (ALE) Kähler manifold, assuming only the sort of weak fall-off conditions required for the mass to actually be well-defined. For ALE scalar-flat Kähler manifolds, the mass turns out to be a topological invariant, depending only on the underlying smooth manifold, the first Chern class of the complex structure, and the Kähler class of the metric. When the metric is actually AE (asymptotically Euclidean), our formula not only implies a positive mass theorem for Kähler metrics, but also yields a Penrose-type inequality for the mass.     

Harmonic maps and stability on locally conformal Kähler manifolds

We study harmonic and pluriharmonic maps on locally conformal Kähler manifolds. We prove that there are no nonconstant holomorphic pluriharmonic maps from a locally conformal Kähler manifold to a Kähler manifold and that any holomorphic harmonic map from a compact locally conformal Kähler manifold to a Kähler manifold is stable.     

On the curvature of Kähler–Norden manifolds

Using the one-to-one correspondence between Kähler–Norden and holomorphic Riemannian metrics, important relations between various Riemannian invariants of manifolds endowed with such metrics are established. Especially, the holomorphic versions of the recurrence of the Riemann, Ricci, projective are defined and investigated. For four-dimensional Kähler–Norden manifolds, it is proved that they are of holomorphically recurrent curvature on the set where the holomorphic scalar curvature does not vanish. Furthermore, a four-dimensional Kähler–Norden manifold is (locally) conformally flat if and only if its holomorphic scalar curvature is constant pure imaginary. The present paper continues author’s investigations of Kähler–Norden manifolds from the papers [K. Słuka, On Kähler manifolds with Norden metrics, An. Ştiint. Univ. Al.I. Cuza IaşI Ser. Ia Mat. 47 (2001) 105–122; K. Słuka, Properties of the Weyl conformal curvature of Kähler–Norden manifolds, in: Proc. Colloq. Diff. Geom. on Steps in Differential Geometry, July 25–30, 2000, Debrecen, 2001, pp. 317–328].     

Cohomological Yang–Mills theories on Kähler 3-folds

We study topological gauge theories with N_c=(2,0) supersymmetry based on stable bundles on general Kähler 3-folds. In order to have a theory that is well defined and well behaved, we consider a model based on an extension of the usual holomorphic bundle by including a holomorphic 3-form. The correlation functions of the model describe complex 3-dimensional generalizations of Donaldson–Witten type invariants. We show that the path integral can be written as a sum of contributions from stable bundles and a complex 3-dimensional version of Seiberg–Witten monopoles. We study certain deformations of the theory, which allow us to consider the situation of reducible connections. We shortly discuss situations of reduced holonomy. After dimensional reduction to a Kähler 2-fold, the theory reduces to Vafa–Witten theory. On a Calabi–Yau 3-fold, the supersymmetry is enhanced to N_c=(2,2). This model may be used to describe classical limits of certain compactifications of (matrix) string theory.     

A new visualization of the gauged supersymmetric sigma-model by Killing potentials

The differential geometry of Kähler group manifolds will be thoroughly interpreted through Killing potentials. This enables us to reformulate the four dimensional gauged supersymmetric σ-model on Kähler group manifolds by Killing potentials. In the reformulation the Lagrangian will take a simple form in which the isometry of the manifolds is linearly manifest. The scalar curvature of the manifolds will be ascribed to the spontaneous breaking of supersymmetry in the model.     

ASK/PSK-correspondence and the <Emphasis Type="Italic">r</Emphasis>-map

We formulate a correspondence between affine and projective special Kähler manifolds of the same dimension. As an application, we show that, under this correspondence, the affine special Kähler manifolds in the image of the rigid r-map are mapped to one-parameter deformations of projective special Kähler manifolds in the image of the supergravity r-map. The above one-parameter deformations are interpreted as perturbative \(\alpha '\)-corrections in heterotic and type II string compactifications with \(N=2\) supersymmetry. Also affine special Kähler manifolds with quadratic prepotential are mapped to one-parameter families of projective special Kähler manifolds with quadratic prepotential. We show that the completeness of the deformed supergravity r-map metric depends solely on the (well-understood) completeness of the undeformed metric and the sign of the deformation parameter.     

Higgs bundles and four manifolds

It is known that the Seiberg–Witten invariants, derived from supersymmetric Yang–Mill theories in four dimensions, do not distinguish smooth structure of certain non-simply-connected four manifolds. We propose generalizations of Donaldson–Witten and Vafa–Witten theories on a Kähler manifold based on Higgs bundles. We showed, in particular, that the partition function of our generalized Vafa–Witten theory can be written as the sum of contributions our generalized Donaldson–Witten invariants and generalized Seiberg–Witten invariants. The resulting generalized Seiberg–Witten invariants might have, conjecturally, information on smooth structure beyond the original Seiberg–Witten invariants for non-simply-connected case.     

Completeness in Supergravity Constructions

We prove that the supergravity r- and c-maps preserve completeness. As a consequence, any component \({\mathcal{H}}\) of a hypersurface {h = 1} defined by a homogeneous cubic polynomial h such that \({-\partial^2h}\) is a complete Riemannian metric on \({\mathcal{H}}\) defines a complete projective special Kähler manifold and any complete projective special Kähler manifold defines a complete quaternionic Kähler manifold of negative scalar curvature. We classify all complete quaternionic Kähler manifolds of dimension less or equal to 12 which are obtained in this way and describe some complete examples in 16 dimensions.     

Superconformal Structures on Generalized Calabi-Yau Metric Manifolds

We construct an embedding of two commuting copies of the N = 2 superconformal vertex algebra in the space of global sections of the twisted chiral-anti-chiral de Rham complex of a generalized Calabi-Yau metric manifold, including the case when there is a non-trivial H-flux and non-vanishing dilaton. The 4 corresponding BRST charges are well defined on any generalized Kähler manifold. This allows one to consider the half-twisted model defining thus the chiral de Rham complex of a generalized Kähler manifold. The classical limit of this result allows one to recover the celebrated generalized Kähler identities as the degree zero part of an infinite dimensional Lie superalgebra attached to any generalized Kähler manifold. As a byproduct of our study we investigate the properties of generalized Calabi-Yau metric manifolds in the Lie algebroid setting.     

Quaternionic Kähler Detour Complexes and $${\mathcal{N} = 2}$$ Supersymmetric Black Holes

We study a class of supersymmetric spinning particle models derived from the radial quantization of stationary, spherically symmetric black holes of four dimensional \({{\mathcal N} = 2}\) supergravities. By virtue of the c-map, these spinning particles move in quaternionic Kähler manifolds. Their spinning degrees of freedom describe mini-superspace-reduced supergravity fermions. We quantize these models using BRST detour complex technology. The construction of a nilpotent BRST charge is achieved by using local (worldline) supersymmetry ghosts to generate special holonomy transformations. (An interesting byproduct of the construction is a novel Dirac operator on the superghost extended Hilbert space.) The resulting quantized models are gauge invariant field theories with fields equaling sections of special quaternionic vector bundles. They underly and generalize the quaternionic version of Dolbeault cohomology discovered by Baston. In fact, Baston’s complex is related to the BPS sector of the models we write down. Our results rely on a calculus of operators on quaternionic Kähler manifolds that follows from BRST machinery, and although directly motivated by black hole physics, can be broadly applied to any model relying on quaternionic geometry.     

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.     

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.     

Instanton-like solutions in chiral models

General two-dimensional Euclidean chiral models of field theory are considered in detail. It is shown that in the case when the field takes its values in an arbitrary Kähler manifold the “duality equations” reduce to the Cauchy- Riemann equations on this manifold. For homogeneous manifolds the solutions of these equations do exist and are given by rational functions.     

Inflation and integrable one‐field cosmologies embedded in rheonomic supergravity

In this paper we show that the new approach to the embedding of the inflationary potentials into supergravity, presented in a quite recent paper [11] of Ferrara, Kallosh, Linde and Porrati can be formulated within the framework of standard matter coupled supergravity, without the use of the new minimal auxiliary set and of conformal compensators. The only condition is the existence of a translational Peccei Quinn isometry of the scalar Kähler manifold. We suggest that this embedding strategy based on a nilpotent gauging amounts to a profound Copernican Revolution. The properties of the inflaton potential are encoded in the geometry of some homogeneous one‐dimensional Kähler manifolds that now should be regarded as the primary object, possibly providing a link with microscopic physics. We present a simple and elegant formula for the curvature of the Kähler manifold in terms of the potential. Most relevant consequence of the new strategy is that all the integrable potentials quite recently classified in a paper [7] that we have coauthored, are automatically embedded into supergravity and their associated Kähler manifolds demand urgent study. In particular one integrable potential that provides the best fit to PLANCK data seems to have inspiring geometrical properties deserving further study.     

The 2-dimensionalN=2 supersymmetric Yang-Mills model on the discrete space-time lattice

We use the Dirac-Kähler formalism in order to put a two-dimensional Yang-Mills theory with a complex supersymmetry on the Euclidean space-time lattice. The theory presents two supersymmetric invariant actions related to each other by a kind of “time reflection” transformation.     

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.     

Observables in topological Yang-Mills theorieswith extended shift supersymmetry

We present a complete classification, at the classical level, of the observables of topological Yang-Mills theories with an extended shift supersymmetry of N generators, in any space-time dimension. The observables are defined as the Yang-Mills BRST cohomology classes of shift supersymmetry invariants. These cohomology classes turn out to be solutions of an N-extension of Witten's equivariant cohomology. This work generalizes results known in the case of shift supersymmetry with a single generator. Received: 8 March 2005, Published online: 21 October 2005 Supported in part by the Conselho Nacional de Desenvolvimento Científico e Tecnológico CNPq, Brazil     

On the Hyperkähler/Quaternion Kähler Correspondence

A hyperkähler manifold with a circle action fixing just one complex structure admits a natural hyperholomorphic line bundle. This observation forms the basis for the construction of a corresponding quaternionic Kähler manifold in the work of A.Haydys. In this paper the corresponding holomorphic line bundle on twistor space is described and many examples computed, including monopole and Higgs bundle moduli spaces. Finally a twistor version of the hyperkähler/quaternion Kähler correspondence is established.     

Reduction of Vaisman structures in complex and quaternionic geometry

We consider locally conformal Kähler geometry as an equivariant (homothetic) Kähler geometry: a locally conformal Kähler manifold is, up to equivalence, a pair $(K,\Gamma )$, where $K$ is a Kähler manifold and $\Gamma$ is a discrete Lie group of biholomorphic homotheties acting freely and properly discontinuously. We define a new invariant of a locally conformal Kähler manifold $(K,\Gamma )$ as the rank of a natural quotient of $\Gamma$, and prove its invariance under reduction. This equivariant point of view leads to a proof that locally conformal Kähler reduction of compact Vaisman manifolds produces Vaisman manifolds and is equivalent to a Sasakian reduction. Moreover, we define locally conformal hyperKähler reduction as an equivariant version of hyperKähler reduction and in the compact case we show its equivalence with 3-Sasakian reduction. Finally, we show that locally conformal hyperKähler reduction induces hyperKähler with torsion (HKT) reduction of the associated HKT structure and the two reductions are compatible, even though not every HKT reduction comes from a locally conformal hyperKähler reduction.