Generalized holomorphic structures

We investigate generalized holomorphic structures in generalized complex geometry. We find that a generalized holomorphic vector bundle carries a generalized complex structure on its total space if some additional conditions hold. We prove that generalized holomorphicity is equivalent to the integrability of a distribution on the total space, and a family of linear Dirac structures associated with this distribution is a generalized complex structure if a further condition holds. Under the same condition, we also prove that local generalized holomorphic frames exist around a regular point.     

Generalized holomorphic bundles and the B-field action

On a generalized complex manifold, there is an associated definition of a generalized holomorphic bundle, introduced by Gualtieri. In the case of an ordinary complex structure, this notion yields an object which we call a co-Higgs bundle, and we consider the B-field action of a closed form of type (1,1)$(1,1)$, both local and global. The effect makes contact with both Nahm’s equations and holomorphic gerbes.     

Uniqueness of balanced metrics on holomorphic vector bundles

Let E→M$E\to M$ be a holomorphic vector bundle over a compact Kähler manifold (M,ω)$(M,\omega )$. We prove that if E$E$ admits a ω$\omega$-balanced metric (in X. Wang’s terminology (Wang, 2005 [3])) then it is unique. This result together with Biliotti and Ghigi (2008) [14] implies the existence and uniqueness of ω$\omega$-balanced metrics of certain direct sums of irreducible homogeneous vector bundles over rational homogeneous varieties. We finally apply our result to show the rigidity of ω$\omega$-balanced Kähler maps into Grassmannians.     

An invariant formula for a star product with separation of variables

We give an invariant formula for a star product with separation of variables on a pseudo-Kähler manifold.     

On coherent states for the simplest quantum groups

The coherent states for the simplest quantum groups (q-Heisenberg-Weyl, SU _q (2) and the discrete series of representations of SU _q (1, 1)) are introduced and their properties investigated. The corresponding analytic representations, path integrals, and q-deformation of Berezin's quantization on , a sphere, and the Lobatchevsky plane are discussed.     

On the parallel transport of the Ricci curvatures

Geometrical characterizations are given for the tensor R⋅S$R\cdot S$, where S$S$ is the Ricci tensor   of a (semi-)Riemannian manifold (M,g)$(M,g)$ and R$R$ denotes the curvature operator   acting on S$S$ as a derivation, and of the Ricci Tachibana tensor  _∧g⋅S${\wedge }_g\cdot S$, where the natural metrical operator  _∧g${\wedge }_g$ also acts as a derivation on S$S$. As a combination, the Ricci curvatures   associated with directions on M$M$, of which the isotropy determines that M$M$ is Einstein, are extended to the Ricci curvatures of Deszcz   associated with directions and planes on M$M$, and of which the isotropy determines that M$M$ is Ricci pseudo-symmetric in the sense of Deszcz.     

A connection with parallel torsion on almost hypercomplex manifolds with Hermitian and anti-Hermitian metrics

Almost hypercomplex manifolds with Hermitian and anti-Hermitian metrics are considered. A linear connection D$D$ is introduced such that the structure of these manifolds is parallel with respect to D$D$. Of special interest is the class of the locally conformally equivalent manifolds of the manifolds with covariantly constant almost complex structures and the case when the torsion of D$D$ is D$D$-parallel. Curvature properties of these manifolds are studied. An example of 4-dimensional manifolds in the considered basic class is constructed and characterized.     

Generalized complex structures on Kodaira surfaces

We compute the deformations in the sense of generalized complex structures of the standard classical complex structure on a primary Kodaira surface and we prove that the obtained family of deformations is a smooth locally complete family depending on four complex parameters. This family is the same as the extended deformations (in the sense of Kontsevich and Barannikov) in degree two, obtained by Poon using differential Gerstenhaber algebras.     

Axiomatic holonomy maps and generalized Yang-Mills moduli space

This Letter is a follow-up of Barrett, J. W.,Internat. J. Theoret. Phys. 30(9), (1991). Its main goal is to provide an alternative proof of that part of the reconstruction theorem which concerns the existence of a connection. A construction of a connection 1-form is presented. The formula expressing the local coefficients of the connection in terms of the holonomy map is obtained as an immediate consequence of that construction. Thus, the derived formula coincides with that used in Chan, H.-M., Scharbach, P., and Tsou, S. T.,Ann. Physics 166, 396–421 (1986). The reconstruction and representation theorems form a generalization of the fact that the pointed configuration space of the classical Yang-Mills theory is equivalent to the set of all holonomy maps. The point of this generalization is that there is a one-to-one correspondence not only between the holonomy maps and the orbits in the space of connections, but also between all maps M G fulfilling some axioms and all possible equivalence classes ofP(M, G) bundles with connections, where the equivalence relation is defined by a bundle isomorphism in a natural way.     

Einstein gravity in almost Kähler and Lagrange–Finsler variables and deformation quantization

A geometric procedure is elaborated for transforming (pseudo) Riemannian metrics and connections into canonical geometric objects (metric and nonlinear and linear connections) for effective Lagrange, or Finsler, geometries which, in turn, can be equivalently represented as almost Kähler spaces. This allows us to formulate an approach to quantum gravity following standard methods of deformation quantization. Such constructions are performed not on tangent bundles, as in usual Finsler geometry, but on spacetimes enabled with nonholonomic distributions defining 2+2$2+2$ splitting with associate nonlinear connection structure. We also show how the Einstein equations can be written in terms of Lagrange–Finsler variables and corresponding almost symplectic structures and encoded into the zero-degree cohomology coefficient for a quantum model of Einstein manifolds.     

An integration formula for the square of moment maps of circle actions

The integration of the exponential of the square of the moment map of the circle action is studied by a direct stationary phase computation and by applying the Duistermaat-Heckman formula. Both methods yield two distinct formulas expressing the integral in terms of contributions from the critical set of the square of the moment map. Certain cohomological pairings on the symplectic quotient are computed explicitly using the asymptotic behavior of the two formulas.     

Orderings and Non-formal Deformation Quantization

We propose suitable ideas for non-formal deformation quantization of Fréchet Poisson algebras. To deal with the convergence problem of deformation quantization, we employ Fréchet algebras originally given by Gel’fand–Shilov. Ideas from deformation quantization are applied to expressions of elements of abstract algebras, which leads to a notion of “independence of ordering principle”. This principle is useful for the understanding of the star exponential functions and for the transcendental calculus in non-formal deformation quantization. Akira Yoshioka was partially supported by Grant-in-Aid for Scientific Research (#19540103.), Ministry of Education, Science and Culture, Japan.     

Deformation quantization of nonholonomic almost Kähler models and Einstein gravity

Nonholonomic distributions and adapted frame structures on (pseudo) Riemannian manifolds of even dimension are employed to build structures equivalent to almost Kähler geometry and which allows to perform a Fedosov-like quantization of gravity. The nonlinear connection formalism that was formally elaborated for Lagrange and Finsler geometry is implemented in classical and quantum Einstein gravity.     

Scalar curvature and holomorphy potentials

A holomorphy potential is a complex valued function whose complex gradient, with respect to some Kähler metric, is a holomorphic vector field. Given k$k$ holomorphic vector fields on a compact complex manifold, form, for a given Kähler metric, a product of the following type: a function of the scalar curvature multiplied by functions of the holomorphy potentials of each of the vector fields. It is shown that the stipulation that such a product be itself a holomorphy potential for yet another vector field singles out critical metrics for a particular functional. This may be regarded as a generalization of the extremal metric variation of Calabi, where k=0$k=0$ and the functional is the square of the L²$L^2$-norm of the scalar curvature. The existence question for such metrics is examined in a number of special cases. Examples are constructed in the case of certain multifactored product manifolds. For the SKR metrics investigated by Derdzinski and Maschler and residing in the complex projective space, it is shown that only one type of nontrivial criticality holds in dimension three and above.     

Supersymmetric WZ-Poisson Sigma Model and Twisted Generalized Complex Geometry

It has been shown recently that extended supersymmetry in twisted first-order sigma models is related to twisted generalized complex geometry in the target. In the general case there are additional algebraic and differential conditions relating the twisted generalized complex structure and the geometrical data defining the model. We study in the Hamiltonian formalism the case of vanishing metric, which is the supersymmetric version of the WZ-Poisson sigma model. We prove that the compatibility conditions reduce to an algebraic equation, which represents a considerable simplification with respect to the general case. We also show that this algebraic condition has a very natural geometrical interpretation. In the derivation of these results the notion of contravariant connections on twisted Poisson manifolds turns out to be very useful.     

Geometry of projective plane and Poisson structure

V.I. Arnold [V. I. Arnold, Lobachevsky triangle altitudes theorem as the Jacobi identity in the Lie algebra of quadratic forms on symplectic plane, Journal of Geometry and Physics, 53 (4) (2005), 421–427] gave an alternative proof to the Lobachevsky triangle altitudes theorem by using a Poisson bracket for quadratic forms and its Jacobi identity, and showed that the orthocenter theorem can be extended on RP²$\mathbb{R}{P}^2$. In this paper, we find a new identity in the Poisson algebra of quadratic forms. Following Arnold’s idea, the goal of this article is to give alternative proofs to theorems, of Desargues, Pascal, and Brianchon, in RP²$\mathbb{R}{P}^2$, by using the Poisson bracket and the identity.