Stable forms and special metrics
By N. Hitchin. 2001
454 citations recorded by the source corpus.
We show how certain diffeomorphism-invariant functionals on differential forms in dimensions 6,7 and 8 generate in a natural way special geometrical structures in these dimensions: metrics of holonomy G2 and Spin(7), metrics with weak holonomy SU(3) and G2, and a new and unexplored example in dimension 8. The general formalism becomes a practical tool for calculating homogeneous or cohomogeneity one examples, and we illustrate this with some newly discovered cases of Spin(7) and G2 metrics.
Subfields: Applied Mathematics, Astronomy and Astrophysics, Geometry and Topology
View Paper: https://www.semanticscholar.org/paper/12e77ca5e51ef0b6a5e6c7f3f5f538335f5cf41b
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Citations in the corpus
- Generalized Killing spinors in dimension 5 (2005, methodology): “In dimensions 6 and 7, Hitchin has translated the embedding property into a problem of existence of integral lines for a vector field on an infinite-dimensional space [19].”
- Flux compactifications in string theory: A Comprehensive review (2005, methodology): “Hitchin showed [96, 124] that there is a Special Kähler structure on the space of generalized almost complex structures (for usual almost complex structures, this bundle is known as the twistor bundle [125]).”
- Six-Dimensional Nearly Kähler Manifolds of Cohomogeneity One (II) (2009, background): “We start from the following known fact: any strict NK structure (g, J ) on a given six dimensional, oriented manifold can be completely recovered by its Kähler form ω = g(J ·, ·) and any non-degenerate 2-form ω, whose differential dω is stable in the sense of Hitchin ([19,20]) and satisfying a suita”
- Geometric Model for Complex Non-Kähler Manifolds with SU (3) Structure (2002, background): “These are complex half-flat manifolds [20] which are upliftable to G2 spaces [27].”
- Kähler Reduction of Metrics with Holonomy G2 (2003, methodology): “Elements of V represent deformations of φ + in a fixed cohomology class, and those of V ∗ deformations of ρ 2 . Given a solution of (32) for τ = a, a solution of (33) can then be found on some interval (a, b) [ 21 ].”
- Hitchin functionals in N = 2 supergravity (2005, background): “[51, 52 , 48], that it admits a special K¨ahler geometry with a K¨ahler potential given by a Hitchin functional.”
- Topological M-theory as unification of form theories of gravity (2004, methodology): “We will describe actions constructed by Hitchin [10,42] for which the equations of motion yield special geometric structures: either holomorphic 3-forms Ω and symplectic structures k in 6 dimensions, or G2 holonomy metrics in 7 dimensions.”
- SU(3) × SU(3) compactification and mirror duals of magnetic fluxes (2006, background): “he deformations of the complex structure [72]. B.2 Truncation conditions The untruncated theory Let us now review how special K¨ahler manifolds appear in the context of generalised geometry following [2, 3, 4] (see also [45] for a review). Let S±(E) be the positive and negative chirality generalised”
- Remarks on the geometry of almost complex 6-manifolds (2005, background): “These forms are the sections of an open subset of A(3)(TM) and hence are stable in Hitchin’s sense [16].”
- Generalized N=1 orientifold compactifications and the Hitchin functionals (2006, methodology): “This condition can also be formulated in terms of the map q : Λ(3)T ∗ → Λ(6)T ∗ ⊗ Λ(6)T ∗ defined as [75]”
- Cohomogeneity One Manifolds of Spin(7) and G2 Holonomy (2001, methodology): “Some properties of the Atiyah-Hitchin solutions are reviewed in Appendix B.”
- New invariants of G2–structures (2012, unknown): “One can attempt to construct Spin(7) metrics on M × (−ǫ, ǫ) using the ‘Hitchin flow’ of coclosed G2-structures [20].”