Special Lagrangian manifolds obtained from complex Grassmannians (original) (raw)
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In this paper we begin the development of a relative version of T-duality in generalized complex geometry which we propose as a manifestation of mirror symmetry. Let M be an n-dimensional smooth real manifold, V a rank n real vector bundle on M, and ∇ a flat connection on V. We define the notion of a ∇-semi-flat generalized almost complex structure on the total space of V. We show that there is an explicit bijective correspondence between ∇-semi-flat generalized almost complex structures on the total space of V and ∇ ∨-semi-flat generalized almost complex structures on the total space of V ∨. We show that semi-flat generalized complex structures give rise to a pair of transverse Dirac structures on the base manifold. We also study the ways in which our results generalize some aspects of T-duality such as the Buscher rules. We show explicitly how spinors are transformed and discuss the induces correspondence on branes under certain conditions.
On Homogeneous Hypersurfaces in Complex Grassmannians
2002
Abstract. In the present article we consider a class of real hypersurfaces of theGrassmann manifold of k-planes in C n , G k (C n ), for k > 2. Namely the familyof tubes around G k (C m ) with m < n and around the quaternionic Grassmannmanifold of k/2-quaternionic planes in H n/2 , G k/2 (H n/2 ), when k and n are even.We determine which of those tubes are homogeneous and for them we find thespectral decomposition of the shape operator. As a consequence we show that theyare Hopf hypersurfaces.MSC 2000: 53C30, 53C35, 53C42Keywords: complex Grassmannians, real hypersurfaces, tubes, shape operator,Kaehler structure1. IntroductionThe study of real hypersurfaces in complex projective spaces has a long and interestinghistory. A nice survey can be found in [11]. A particular subclass is that of the homogeneoushypersurfaces which at the present time seems to be very well understood. In this respectthe reader may get well acquainted with this topic by reading the article [8] and referen...
Grassmannians, calibrations and 5-brane intersections
Classical and Quantum Gravity, 2000
We present a geometric construction of a new class of hyper-Kähler manifolds with torsion. This involves the superposition of the four-dimensional hyper-Kähler geometry with torsion associated with the NS-5-brane along quaternionic planes in H k . We find the moduli space of these geometries and show that it can be constructed using the bundle space of the canonical quaternionic line bundle over a quaternionic projective space. We also investigate several special cases which are associated with certain classes of quaternionic planes in H k . We then show that the eight-dimensional geometries we have found can be constructed using quaternionic calibrations. We generalize our construction to superpose the same four-dimensional hyper-Kähler geometry with torsion along complex planes in C 2k .
2018
The aim of these notes is to describe how to construct canonical bundles of moving frames and differential invariants for parametrized curves in Lagrangian Grassmannians, at least in the monotonic case. Such curves appear as Jacobi curves of sub-Riemannian extremals [1,2] . Originally this construction was done in [6,7], where it uses the specifics of Lagrangian Grassmannian. In later works [3,4] a much more general theory for construction of canonical bundles of moving frames for parametrized or unparametrized curves in the so-called generalized flag varieties was developed, so that the problem which is discussed here can be considered as a particular case of this general theory. Although this was briefly discussed at the very end of [3], the application of the theory of [3,4] to obtain the results of [6,7] were never written in detail and this is our goal here. We believe that this exposition gives a more conceptual point of view on the original results of [6,7] and especially cla...
Mirror duality and G(2) manifolds
2006
The main purpose of this note is to give a mathematical definition of the " mirror symmetry " and discuss its applications. More specifically we explain how to assign a G2 manifold (M, ϕ, Λ), with the calibration 3-form ϕ and an oriented 2-plane field Λ, a pair of parametrized tangent bundle valued 2 and 3-forms of M. These forms can then be used to define different complex and symplectic structures on certain 6-dimensional subbundles of T (M). When these bundles integrated they give mirror CY manifolds. For example, in the special case of M =Calabi-Yau×S 1 , one of the 6-dimensional subbundles corresponds to the tangent bundle of the CY manifold. This explains the mirror duality between the symplectic and complex structures on the CY 3-folds inside of a G2 manifold. One can extend these arguments to noncompact G2 manifolds of the form CY×R.
A generalized construction of mirror manifolds
Nuclear Physics B, 1993
We generalize the known method for explicit construction of mirror pairs of (2, 2)-superconformal field theories, using the formalism of Landau-Ginzburg orbifolds. Geometrically, these theories are realized as Calabi-Yau hypersurfaces in weighted projective spaces. This generalization makes it possible to construct the mirror partners of many manifolds for which the mirror was not previously known.
Calibrated geometries in Grassmann manifolds
Commentarii Mathematici Helvetici, 1989
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Differential Geometry Parametrized curves in Lagrange Grassmannians
2007
Curves in Lagrange Grassmannians naturally appear when one studies Jacobi equations for extremals, associated with geometric structures on manifolds. We fix integers di and consider curves Λ(t) for which at each t the derivatives of order ≤ i of all curves of vectors `(t) ∈ Λ(t) span a subspace of dimension di. We will describe the construction of a complete system of symplectic invariants for such parametrized curves, satisfying a certain genericity assumption, and give applications to geometric structures, including sub-Riemannian and sub-Finslerian