Generalized Calabi-Yau manifolds

By N. Hitchin. 2002

1,411 citations recorded by the source corpus.

A geometrical structure on even-dimensional manifolds is defined which generalizes the notion of a Calabi–Yau manifold and also a symplectic manifold. Such structures are of either odd or even type and can be transformed by the action both of diffeomorphisms and closed 2-forms. In the special case of six dimensions we characterize them as critical points of a natural variational problem on closed forms, and prove that a local moduli space is provided by an open set in either the odd or even cohomology. We introduce in this paper a geometrical structure on a manifold which generalizes both the concept of a Calabi–Yau manifold—a complex manifold with trivial canonical bundle—and that of a symplectic manifold. This is possibly a useful setting for the background geometry of recent developments in string theory; but this was not the original motivation for the author’s first encounter with this structure. It arose instead as part of a programme (following the papers [ 11, 12]) for characterizing special geometry in low dimensions by means of invariant functionals of differential forms. In this respect, the dimension six is particularly important. This paper has two aims, then: first to introduce the general concept, and then to look at the variational and moduli space problem in the special case of six dimensions. We begin with the definition in all dimensions of what we call generalized complex manifolds and generalized Calabi–Yau manifolds .

Subfields: Applied Mathematics, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/e24049a48784431de621b46002c7af7da429c331

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Applied Mathematics

In Geometry and Topology

In Mathematical Physics

Citations in the corpus

The decomposition of forms and cohomology of generalized complex manifolds (2006, positive): “In this paper we investigate properties similar to the ones above for generalized complex manifolds, as introduced by Nigel Hitchin [8] and studied by Marco Gualtieri [6].”
New aspects of the ddc-lemma (2005, positive): “Also, as early examples show [44], in some cases it is possible to deform complex to symplectic structures on a manifold by a path of generalized complex structures.”
Mirror symmetry in generalized Calabi–Yau compactifications ☆ (2002, background): “that the manifold after the ‘topology-changing’ transition is still complex but non-K¨ahler. (In our terminology these manifolds correspond to W1 = W2 = 0.) 19In this respect a generalization of ref. [73] might be useful. 29 superpotential is given by [31,32] WRR A = Z Y F6 +J∧ F4 +J2 ∧ F2 +J3F0 . (”
Flux compactifications in string theory: A Comprehensive review (2005, background): “This description involves generalized complex geometry [95, 96].”
Cohomology and Hodge Theory on Symplectic Manifolds: III (2009, background): “Having noted the similarities of cohomologies in complex and symplectic geometries, it is natural to consider extending the cohomologies to the generalized complex geometries introduced by Hitchin [11] (see also [8, 5]).”
A scan for new N = 1 vacua on twisted tori (2006, background): “The differential equations that supersymmetry imposes on them are best analyzed in the framework of generalized complex geometry (GCG) [17,18], which was developed concurrently with the progress on the physics side of the problem (an applied review of GCG will be given in Section 3).”
Generalized complex structures on nilmanifolds (2004, background): “[3] Hitchin, N. Generalized Calabi-Yau Manifolds.”
Asymmetric orbifolds, non-geometric fluxes and non-commutativity in closed string theory (2012, methodology): “We mention here the approach double field theory [14] which takes T-duality as a fundamental symmetry of the theory and the one of generalized geometry [34–37] which attempts to generalize classical geometry by including the B-field as a gerbe connection on a formal sum of the tangent and cotangent ”
A surgery for generalized complex structures on 4-manifolds (2006, background): “We also thank Nigel Hitchin for helpful conversations.”
Generalized metric formulation of double field theory (2010, background): “In the mathematical literature, the generalized metric and the Courant bracket are key structures in generalized geometry [15, 16, 17].”
Submanifolds of generalized complex manifolds (2003, background): “The definition of a GCS (Generalized Complex Structure) in terms of the subspace E ⊆ VC ⊕V ∗ C first appeared in [7], the definition in terms of an R-linear automorphism J of V ⊕V ∗ was introduced in [4], and the definition in terms of spinors appears implicitly in [7] and explicitly in [8].”
Reduction and submanifolds of generalized complex manifolds (2005, methodology): “This notion is based on Hitchin’s remark [10] that, for any closed 2-form B on M , the mapping”
Supersymmetric backgrounds from generalized Calabi‐Yau manifolds (2004, background): “A manifold with SU(3)-structure has all the group-theoretical features of a Calabi–Yau, namely invariant two- and three forms, J and Ω respectively.”
The Gauge algebra of double field theory and Courant brackets (2009, background): “In §7 we show that for parameters restricted to be independent of x̃i, the C-bracket is precisely the Courant bracket [6], a central construction in generalised geometry [7, 8, 9].”
Equivalences of twisted K3 surfaces (2004, methodology): “This notion was introduced by Hitchin in [16], further discussed in Gualtieri’s theses [15] and in the case of K3 surfaces in [20].”
From ten to four and back again: how to generalize the geometry (2007, background): “Furthermore, let us keep in mind that, as shown in [5], the complex pure spinors are in fact completely defined by their real or imaginary part.”