Kaehler Geometry Research Articles

Geometry of CR-slant Warped Products in Nearly Trans-Sasakian Manifolds

A CR-slant warped product is considered a generalization of a CR-warped product. It was firstly introduced and studied in the field of Kaehler geometry. In this paper, we study CR-slant warped products in nearly trans-Sasakian manifolds. We obtain a general inequality related to the second fundamental form and the warping function in such warped products. The equality case is discussed.

Differentialgeometrie im Großen

The topics discussed at the meeting were Kaehler geometry, geometric evolution equations, manifolds of nonnegative curvature, metric geometry and geometric representations of groups. The choice of topics reflects current trends in the development of differential geometry.

Strongly Hermitian Einstein–Maxwell solutions on ruled surfaces

This paper produces explicit strongly Hermitian Einstein–Maxwell solutions on the smooth compact 4-manifolds that are $$S^2$$ -bundles over compact Riemann surfaces of any genus. This generalizes the existence results by LeBrun (J Geom Phys 91:163–171, 2015, The Einstein–Maxwell equations and conformally Kaehler geometry. arXiv:1504.06669 , 2016). Moreover, by calculating the (normalized) Einstein–Hilbert functional of our examples, we generalize Theorem E of LeBrun (The Einstein–Maxwell equations and conformally Kaehler geometry. arXiv:1504.06669 , 2016), which speaks to the abundance of Hermitian Einstein–Maxwell solutions on such manifolds. As a bonus, we exhibit certain pairs of strongly Hermitian Einstein–Maxwell solutions, first found in LeBrun (The Einstein–Maxwell equations and conformally Kaehler geometry. arXiv:1504.06669 , 2016), on the first Hirzebruch surface in a form which clearly shows that they are conformal to a common Kahler metric. In particular, this yields a non-trivial example of non-uniqueness of positive constant scalar curvature metrics in a given conformal class.

Black holes and groups of type E 7

We report some results on the relation between extremal black holes in locally supersymmetric theories of gravity and groups of type E7, appearing as generalized electric-magnetic duality symmetries in such theories. Some basics on the covariant approach to the stratification of the relevant symplectic representation are reviewed, along with a connection between special Kaehler geometry and a "generalization" of groups of type E7.

Quantum theory of massless (p, 0)-forms

We describe the quantum theory of massless (p,0)-forms that satisfy a suitable holomorphic generalization of the free Maxwell equations on Kaehler spaces. These equations arise by first-quantizing a spinning particle with a U(1)-extended local supersymmetry on the worldline. Dirac quantization of the spinning particle produces a physical Hilbert space made up of (p,0)-forms that satisfy holomorphic Maxwell equations coupled to the background Kaehler geometry, containing in particular a charge that measures the amount of coupling to the U(1) part of the U(d) holonomy group of the d-dimensional Kaehler space. The relevant differential operators appearing in these equations are a twisted exterior holomorphic derivative and its hermitian conjugate (twisted Dolbeault operators with charge q). The particle model is used to obtain a worldline representation of the one-loop effective action of the (p,0)-forms. This representation allows to compute the first few heat kernel coefficients contained in the local expansion of the effective action and to derive duality relations between (p,0) and (d-p-2,0)-forms that include a topological mismatch appearing at one-loop.

Generalized Pseudo-Kähler Structures

In the paper we consider pseudo bihermitian structures - a pair of complex structures compatible with a pseudo Riemannian metric. As in the positive definite case we establish its relations with generalized (pseudo) Kaehler geometry and holomorphic Poisson structures. We provide a list of compact complex surfaces which could admit such structure and also examples of bihermitian structures on some of them. We also consider a naturally defined null plane distribution on generalized pseudo Kaehler 4-manifold and show that under a mild restriction it determines an Engel structure.

Examples of non-trivial rank in locally conformal Kähler geometry

We consider locally conformal Kaehler geometry as an equivariant, homothetic Kaehler geometry (K,\Gamma). We show that the de Rham class of the Lee form can be naturally identified with the homomorphism projecting \Gamma to its dilation factors, thus completing the description of locally conformal Kaehler geometry in this equivariant setting. The rank r of a locally conformal Kaehler manifold is the rank of the image of this homomorphism. Using algebraic number theory, we show that r is non-trivial, providing explicit examples of locally conformal Kaehler manifolds with 1<r<b_1. As far as we know, these are the first examples of this kind. Moreover, we prove that locally conformal Kaehler Oeljeklaus-Toma manifolds have either r=b_1 or r=b_1/2.

Jordan frame supergravity and inflation in the NMSSM

We present a complete explicit N=1, d=4 supergravity action in an arbitrary Jordan frame with non-minimal scalar-curvature coupling of the form $\Phi(z, \bar z)\, R$. The action is derived by suitably gauge-fixing the superconformal action. The theory has a modified Kaehler geometry, and it exhibits a significant dependence on the frame function $\Phi (z, \bar z)$ and its derivatives over scalars, in the bosonic as well as in the fermionic part of the action. Under certain simple conditions, the scalar kinetic terms in the Jordan frame have a canonical form. We consider an embedding of the Next-to-Minimal Supersymmetric Standard Model (NMSSM) gauge theory into supergravity, clarifying the Higgs inflation model recently proposed by Einhorn and Jones. We find that the conditions for canonical kinetic terms are satisfied for the NMSSM scalars in the Jordan frame, which leads to a simple action. However, we find that the gauge singlet field experiences a strong tachyonic instability during inflation in this model. Thus, a modification of the model is required to support the Higgs-type inflation.

A Hilbert–Mumford criterion for polystability in Kaehler geometry

Consider a Hamiltonian action of a compact Lie group K on a Kaehler manifold X with moment map μ: X → * . Assume that the action of K extends to a holomorphic action of the complexification G of K. We characterize which G-orbits in X intersect μ ―1 (0) in terms of the maximal weights lim t→∞ 〈μ(e its · x), s), where s ∈ . We do not impose any a priori restriction on the stabilizer of x. Under some mild restrictions on the action K X, we view the maximal weights as defining a collection of maps: for each x ∈ X, λ x : ∂ ∞ (K\G) → ℝ ∪ {∞}, where ∂ ∞ (K\G) is the boundary at infinity of the symmetric space K\G. We prove that G · x ∩ μ ―1 (0) ≠ 0 if: (1) λ x is everywhere nonnegative, (2) any boundary point y such that λ x (y) = 0 can be connected with a geodesic in K\G to another boundary point y' satisfying λ x (y') = 0. We also prove that the maximal weight functions are G-equivariant: for any g ∈ G and any y ∈ ∂ ∞ (K\G) we have λ g·x (y) = λ x (y · g).

A special road to AdS vacua

We apply the techniques of special Kaehler geometry to investigate AdS_4 vacua of general N=2 gauged supergravities underlying flux compactifications of type II theories. We formulate the scalar potential and its extremization conditions in terms of a triplet of prepotentials P_x and their special Kaehler covariant derivatives only, in a form that recalls the potential and the attractor equations of N=2 black holes. We propose a system of first order equations for the P_x which generalize the supersymmetry conditions and yield non-supersymmetric vacua. Special geometry allows us to recast these equations in algebraic form, and we find an infinite class of new N=0 and N=1 AdS_4 solutions, displaying a rich pattern of non-trivial charges associated with NSNS and RR fluxes. Finally, by explicit evaluation of the entropy function on the solutions, we derive a U-duality invariant expression for the cosmological constant and the central charges of the dual CFT's.

Geometry of anN=4twisted string

We compare N=2 string and N=4 topological string within the framework of the sigma model approach. Being classically equivalent on a flat background, the theories are shown to lead to different geometries when put in a curved space. In contrast to the well studied Kaehler geometry characterising the former case, in the latter case a manifold has to admit a covariantly constant holomorphic two-form in order to support an N=4 twisted supersymmetry. This restricts the holonomy group to be a subgroup of SU(1,1) and leads to a Ricci--flat manifold. We speculate that, the N=4 topological formalism is an appropriate framework to smooth down ultraviolet divergences intrinsic to the N=2 theory.

Supersymmetric Quantum Theory and Non-Commutative Geometry

Classical differential geometry can be encoded in spectral data, such as Connes' spectral triples, involving supersymmetry algebras. In this paper, we formulate non-commutative geometry in terms of supersymmetric spectral data. This leads to generalizations of Connes' non-commutative spin geometry encompassing non-commutative Riemannian, symplectic, complex-Hermitian and (Hyper-)Kaehler geometry. A general framework for non-commutative geometry is developed from the point of view of supersymmetry and illustrated in terms of examples. In particular, the non-commutative torus and the non-commutative 3-sphere are studied in some detail.

Fermion masses from supersymmetric dynamics in proper time.

The proper-time superparticle formulation of the Dirac operator is extended by the introduction of supersymmetric dynamics for extra coordinates, representing multiplets of chiral quarks and leptons. The resulting internal Hamiltonian is interpreted as their mass-squared operator. Graded Kaehler geometries are proposed for the additional coordinates, with the {ital a} numbers generating the spectrum of one fermion family, and the {ital c} numbers (in the holomorphic first-order formulation) generating bosonic excitations corresponding to family replication. A simplified SU(5) model is described which is compatible with the SU(5) minimal Higgs scenario (for one standard SU(5) quark and lepton generation plus singlet neutrino).

The axiom of spheres in Kaehler geometry

The axiom of spheres in Kaehler geometry

The axiom of 2-spheres in Kaehler geometry

The axiom of 2-spheres in Kaehler geometry

A generalization of Kaehler geometry

A generalization of Kaehler geometry