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.     

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.     

Hybrid inflation in supergravity with (SU(1,1)/U(1))m Kähler manifolds

In the presence of fields without superpotential but with large vevs through D-terms the mass-squared of the inflaton in the context of supergravity hybrid inflation receives positive contributions which could cancel the possibly negative Kähler potential ones. The mechanism is demonstrated using Kähler potentials associated with products of SU(1,1)/U(1) Kähler manifolds. In a particularly simple model of this type all supergravity corrections to the F-term potential turn out to be proportional to the inflaton mass allowing even for an essentially completely flat inflationary potential. The model also allows for a detectable gravitational wave contribution to the microwave background anisotropy. Its initial conditions are quite natural largely due to a built in mechanism for a first stage of “chaotic” D-term inflation.     

Local index theorem for orbifold Riemann surfaces

We derive a local index theorem in Quillen’s form for families of Cauchy–Riemann operators on orbifold Riemann surfaces (or Riemann orbisurfaces) that are quotients of the hyperbolic plane by the action of cofinite finitely generated Fuchsian groups. Each conical point (or a conjugacy class of primitive elliptic elements in the Fuchsian group) gives rise to an extra term in the local index theorem that is proportional to the symplectic form of a new Kähler metric on the moduli space of Riemann orbisurfaces. We find a simple formula for a local Kähler potential of the elliptic metric and show that when the order of elliptic element becomes large, the elliptic metric converges to the cuspidal one corresponding to a puncture on the orbisurface (or a conjugacy class of primitive parabolic elements). We also give a simple example of a relation between the elliptic metric and special values of Selberg’s zeta function.

     

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.     

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.     

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.     

Homogeneous Kähler manifolds and flat-potential N=1 supergravities

Given a homogeneous Kähler manifold, with trivial Kähler class, there is at least one (usually infinitely many) homogeneous Kähler metric and superpotential such that the scalar potential of the corresponding N=1 SUGRA vanishes identically. We give the explicit form of the function G for the general case.     

Instanton solutions from Abelian sinh-Gordon and Tzitzeica vortices

We study the Abelian Higgs vortex solutions to the sinh-Gordon equation and the elliptic Tzitzeica equation. Starting from these particular vortices, we construct solutions to the Taubes equation with higher vortex number, on surfaces with conical singularities.We then, analyse more general properties of vortices on such singular surfaces and propose a method to obtain vortices on conifolds from vortices on surfaces of revolution. We apply our method to construct explicit vortex solutions on the Poincaré disk with a conical singularity in the centre, to which we refer as the “hyperbolic cone”.We uplift the Abelian sinh-Gordon and Tzitzeica vortex solutions to four dimensions and construct cylindrically symmetric, self-dual Yang–Mills instantons on a non-self-dual (nor anti-self-dual) 4-dimensional Kähler manifold with non-vanishing scalar curvature. The instantons we construct in this way cannot be obtained via a twistorial approach.     

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.     

Holonomy groups,complex structures andD=4 topological Yang-Mills theory

We study conditions for the existence of extended supersymmetry in topological Yang-Mills theory. These conditions are most conveniently formulated in terms of the holonomy group of the underlying manifold, on which the topological Yang-Mills theory is defined. For irreducible manifolds we find that extended supersymmetries are in 1–1 correspondence with covariantly constant complex structures. Therefore, the topological Yang-Mills theory on any Kähler manifold possesses one additional supersymmetry and on any hyper Kähler manifold there are three additional supersymmetries. The Donaldson map, which plays a crucial role in the construction of the topological invariants, is generalized for Kähler manifolds, thus providing candidates for new invariants of complex manifolds.     

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.     

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.     

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.     

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].     

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.     

Gauge Linear Sigma Model for Berglund—Hübsch-Type Calabi—Yau Manifolds

We briefly present the results of our computation of special Kähler geometry for polynomial deformations of Berglund–Hübsch type Calabi–Yau manifolds. We also build mirror symmetric Gauge Linear Sigma Model and check that its partition function computed by supersymmetric localization coincides with exponent of the Kähler potential of the special metric.