Heterotic non-Kähler geometries via polystable bundles on Calabi-Yau threefolds

In arXiv:1008.1018 it is shown that a given stable vector bundle V on a Calabi-Yau threefold X which satisfies c₂(X)=c₂(V) can be deformed to a solution of the Strominger system and the equations of motion of heterotic string theory. In this note we extend this result to the polystable case and construct explicit examples of polystable bundles on elliptically fibered Calabi-Yau threefolds where it applies. The polystable bundle is given by a spectral cover bundle, for the visible sector, and a suitably chosen bundle, for the hidden sector. This provides a new class of heterotic flux compactifications via non-Kähler deformation of Calabi-Yau geometries with polystable bundles. As an application, we obtain examples of non-Kähler deformations of some three generation GUT models.     

A Positive Mass Theorem for Spaces with Asymptotic SUSY Compactification

We prove a positive mass theorem for spaces which asymptotically approach a flat Euclidean space times a Calabi-Yau manifold (or any special honolomy manifold except the quaternionic Kähler). This is motivated by the very recent work of Hertog-Horowitz-Maeda [HHM].     

Hamiltonian constructions of Kähler-Einstein metrics and Kähler metrics of constant scalar curvature

Assuming the existence of a real torus acting through holomorphic isometries on a Kähler manifold, we construct an ansatz for Kähler-Einstein metrics and an ansatz for Kähler metrics with constant scalar curvature. Using this Hamiltonian approach we solve the differential equations in special cases and find, in particular, a family of constant scalar curvature Kähler metrics describing a non-linear superposition of the Bergman metric, the Calabi metric and a higher dimensional generalization of the LeBrun Kähler metric. The superposition contains Kähler-Einstein metrics and all the geometries are complete on the open disk bundle of some line bundle over the complex projective spaceP _n. We also build such Kähler geometries on Kähler quotients of higher cohomogeneity.Partially supported by the NSF Under Grant No. DMS 8906809     

First Eigenvalue of the Dirac Operator for a Kähler Manifold of Even Complex Dimension: The Limiting Case

K. D. Kirchberg has given a minoration of the absolute value of the eigenvalues of the Dirac operator for a compact Kähler spin manifold (W,g) with positive scalar curvature and has introduced, in this context, the notion of Kähler twistor-spinor. We prove here that if dim_C W = p 4 is even, in the limiting case, (W, g) is the Kähler product of an odd-dimensional limiting case compact Kähler spin manifold of complex dimension (p-1), by a flat Kähler manifold which is a compact quotient of C.     

Warped product Kähler manifolds and Bochner–Kähler metrics

Using as an underlying manifold an alpha-Sasakian manifold, we introduce warped product Kähler manifolds. We prove that if the underlying manifold is an alpha-Sasakian space form, then the corresponding Kähler manifold is of quasi-constant holomorphic sectional curvatures with a special distribution. Conversely, we prove that any Kähler manifold of quasi-constant holomorphic sectional curvatures with a special distribution locally has the structure of a warped product Kähler manifold whose base is an alpha-Sasakian space form. As an application, we describe explicitly all Bochner–Kähler metrics of quasi-constant holomorphic sectional curvatures. We find four families of complete metrics of this type. As a consequence, we obtain Bochner–Kähler metrics generated by a potential function of distance in complex Euclidean space and of time-like distance in the flat Kähler–Lorentz space.     

The Dynamics of the Energy of a Kähler Class

The energy of a Kähler class, on a compact complex manifold (M,J) of Kähler type, is the infimum of the squared L²-norm of the scalar curvature over all Kähler metrics representing the class. We study general properties of this functional, and define its gradient flow over all Kähler classes represented by metrics of fixed volume. When besides the trivial holomorphic vector field of (M,J), all others have no zeroes, we extend it to a flow over all cohomology classes of fixed top cup product. We prove that the dynamical system in this space defined by the said flow does not have periodic orbits, that its only fixed points are critical classes of a suitably defined extension of the energy function, and that along solution curves in the Kähler cone the energy is a monotone function. If the Kähler cone is forward invariant under the flow, solutions to the flow equation converge to a critical point of the class energy function. We show that this is always the case when the manifold has a signed first Chern class. We characterize the forward stability of the Kähler cone in terms of the value of a suitable time dependent form over irreducible subvarieties of (M,J). We use this result to draw several geometric conclusions, including the determination of optimal dimension dependent bounds for the squared L²-norm of the scalar curvature functional.Acknowledgement We would like to thank Nicholas Buchdahl for helpful conversations leading us to several improvements of an earlier version of the article, including the correction of two improper assertions.     

La première valeur propre de l'opérateur de Dirac sur les variétés kählériennes compactes

K.D. Kirchberg [Ki1] gave a lower bound for the first eigenvalue of the Dirac operator on a spin compact Kähler manifoldM of odd complex dimension with positive scalar curvature. We prove that manifolds of real dimension 8l+6 satisfying the limiting case are twistor space (cf. [Sa]) of quaternionic Kähler manifold with positive scalar curvature and that the only manifold of real dimension 8l+2 satisfying the limiting case is the complex projective spaceCP ^4l+1.     

Gravitational actions in two dimensions and the Mabuchi functional

The Mabuchi energy is an interesting geometric functional on the space of Kähler metrics that plays a crucial rôle in the study of the geometry of Kähler manifolds. We show that this functional, as well as other related geometric actions, contribute to the effective gravitational action when a massive scalar field is coupled to gravity in two dimensions in a small mass expansion. This yields new theories of two-dimensional quantum gravity generalizing the standard Liouville models.     

Nuclear and string fields on a Kähler manifold

An arbitrary oriented Riemannian manifold of real dimension two is a complex manifold that is also the world sheet of an oriented closed string. Another example is the complex Grassmann manifold ofp-planes inC _2p, which is shown to carry the most symmetric state of⁹Li. In both cases we are concerned with a chiral spinor field on a curved surface that gives rise to anyons in the nuclear case. Specifically we find a distorted torus on a Kähler manifold which is also a Calabi-Yau space.     

An analysis of soft terms in Calabi-Yau compactifications

We perform an analysis of the soft supersymmetry-breaking terms arising in Calabi-Yau compactifications. The sigma-model contribution and the instanton correction to the Kähler potential are included in the computation. The existence of off-diagonal moduli and matter metrics gives rise to specific features as the possibility of having scalars heavier than gauginos or the presence of tachyons. Although non-universal soft terms is a natural situation, we point out that there is an interesting limit where universality is achieved. Finally, we compare these results with those of orbifold compactifications. Although they are qualitatively similar some features indeed change. For example, sum rules found in orbifold models which imply that on average the scalars are lighter than gauginos can be violated in Calabi-Yau manifolds.     

Holomorphic vector fields of compact pseudo-Kähler manifolds

It is well known that a pseudo-Kähler structure is a natural generalization of the Kähler structure. In this paper, we consider holomorphic vector fields of a compact pseudo-Kähler manifold from the viewpoint of Kähler manifolds.     

The effective Kähler potential,metastable vacua and R-symmetry breaking in O'Raifeartaigh models

Much has been learned about metastable vacua and R-symmetry breaking in O'Raifeartaigh models. Such work has largely been done from the perspective of the superpotential and by including Coleman–Weinberg corrections to the scalar potential. Instead, we consider these ideas from the perspective of the one loop effective Kähler potential. We translate known ideas to this framework and construct convenient formulas for computing individual terms in the expanded effective Kähler potential. We do so for arbitrary R-charge assignments and allow for small R-symmetry violating terms so that both spontaneous and explicit R-symmetry breaking is allowed in our analysis.     

HyperKähler and quaternionic Kähler manifolds with S -symmetries

We study relations between quaternionic Riemannian manifolds admitting different types of symmetries. We show that any hyperKähler manifold admitting hyperKähler potential and triholomorphic action of S¹$S^1$ can be constructed from another hyperKähler manifold (of lower dimension) with an action of S¹$S^1$ that fixes one complex structure and rotates the other two and vice versa. We also study the corresponding quaternionic Kähler manifolds equipped with a quaternionic Kähler action of the circle. In particular we show that any positive quaternionic Kähler manifolds with S¹$S^1$-symmetry admits a Kähler metric on an open everywhere dense subset.     

Eingenvalues of the Dirac operator on compact Kähler manifolds

Kählerian twistor operators are introduced to get lower bounds for the eigenvalues of the Dirac operator on compact spin Kähler manifolds. In odd complex dimensions, manifolds with the smallest eigenvalues are characterized by an over determined system of differential equations similar to the Riemannian case. In these dimensions, we show the existence of a unique natural Kählerian twistor operator. It is also proved that, on a Kähler manifold with nonzero scalar curvature, the space of Riemannian twistor-spinors is trivial.This work has been partially supported by the EEC programme GADGET Contract Nr. SC1-0105     

Special Symplectic Connections and Poisson Geometry

By a special symplectic connection we mean a torsion free connection which is either the Levi-Civita connection of a Bochner-Kähler metric of arbitrary signature, a Bochner-bi-Lagrangian connection, a connection of Ricci type or a connection with special symplectic holonomy. A manifold or orbifold with such a connection is called special symplectic. We show that any special symplectic connection can be constructed using symplectic realizations of quadratic deformations of a certain linear Poisson structure. Moreover, we show that these Poisson structures cannot be symplectically integrated by a Hausdorff groupoid. As a consequence, we obtain a canonical principal line bundle over any special symplectic manifold or orbifold, and we deduce numerous global consequences.     

Towards large volume big divisor D3/D7 “μ-split supersymmetry” and Ricci-flat Swiss-cheese metrics, and dimension-six neutrino mass operators

We show that it is possible to realize a “μ-split SUSY” scenario (Cheng and Cheng, 2005) [1] in the context of large volume limit of type IIB compactifications on Swiss-cheese Calabi-Yau orientifolds in the presence of a mobile space-time filling D3-brane and a (stack of) D7-brane(s) wrapping the “big” divisor. For this, we investigate the possibility of getting one Higgs to be light while other to be heavy in addition to a heavy higgsino mass parameter. Further, we examine the existence of long lived gluino that manifests one of the major consequences of μ-split SUSY scenario, by computing its decay width as well as lifetime corresponding to the three-body decays of the gluino into either a quark, a squark and a neutralino or a quark, squark and goldstino, as well as two-body decays of the gluino into either a neutralino and a gluon or a goldstino and a gluon. Guided by the geometric Kähler potential for ΣB obtained in Misra and Shukla (2010) [2] based on GLSM techniques, and the Donaldson?s algorithm (Barun et al., 2008) [3] for obtaining numerically a Ricci-flat metric, we give details of our calculation in Misra and Shukla (2011) [4] pertaining to our proposed metric for the full Swiss-cheese Calabi-Yau (the geometric Kähler potential being needed to be included in the full moduli space Kähler potential in the presence of the mobile space-time filling D3-brane), but for simplicity of calculation, close to the big divisor, which is Ricci-flat in the large volume limit. Also, as an application of the one-loop RG flow solution for the higgsino mass parameter, we show that the contribution to the neutrino masses at the EW scale from dimension-six operators arising from the Kähler potential, is suppressed relative to the Weinberg-type dimension-five operators.     

The effective interactions of chiral families in four-dimensional superstrings

The low-energy effective interactions in four-dimensional superstrings with N = 2 and N = 1 space-time supersymmetry and massless twisted (family) sector are obtained. Our results rely on some general symmetry properties of superstring particle states and on tensor-calculus techniques for supergravity couplings. The novel feature is that the N = 2 quaternionic manifold and N = 1 Kähler space of the scalar superpartners of family multiplets are non-symmetric spaces whose structure can be obtained by “integrating out” the massive superstring modes.     

On reproducing kernels and quantization of states

Quantization of a mechanical system with the phase space a Kähler manifold is studied. It is shown that the calculation of the Feynman path integral for such a system is equivalent to finding the reproducing kernel function. The proposed approach is applied to a scalar massive conformal particle interacting with an external field which is described by deformation of a Hermitian line bundle structure.     

Random Kähler metrics

The purpose of this article is to propose a new method to define and calculate path integrals over metrics on a Kähler manifold. The main idea is to use finite dimensional spaces of Bergman metrics, as an approximation to the full space of Kähler metrics. We use the theory of large deviations to decide when a sequence of probability measures on the spaces of Bergman metrics tends to a limit measure on the space of all Kähler metrics. Several examples are considered.     

de Sitter vacua from matter superpotentials

Consistent uplifting of AdS vacua in string theory often requires extra light degrees of freedom in addition to those of a (Kähler) modulus. Here we consider the possibility that de Sitter and Minkowski vacua arise due to hidden sector matter interactions. We find that, in this scheme, the hierarchically small supersymmetry breaking scale can be explained by the scale of gaugino condensation and that interesting patterns of the soft terms arise. In particular, a matter-dominated supersymmetry breaking scenario and a version of the mirage mediation scheme appear in the framework of spontaneously broken supergravity.