Generalized complex geometry

Abstract

Generalized complex geometry, as developed by Hitchin, contains complex and symplectic geometry as its extremal special cases. In this thesis, we explore novel phenomena exhibited by this geometry, such as the natural action of a B-field. We provide new examples, including some on manifolds admitting no known complex or symplectic structure. We prove a generalized Darboux theorem which yields a local normal form for the geometry. We show that there is an elliptic deformation theory and establish the existence of a Kuranishi moduli space. We then define the concept of a generalized Kahler manifold. We prove that generalized Kahler geometry is equivalent to a bi-Hermitian geometry with torsion first discovered by physicists. We then use this result to solve an outstanding problem in 4-dimensional bi-Hermitian geometry: we prove that there exists a Riemannian metric on the complex projective plane which admits exactly two distinct Hermitian complex structures with equal orientation. Finally, we introduce the concept of generalized complex submanifold, and show that such sub-objects correspond to D-branes in the topological A- and B-models of string theory.

Oxford University DPhil thesis, 107 pages

Cited by in corpus (73)

• Double Field Theory
• Generalized metric formulation of double field theory
• SU(3) x SU(3) compactification and mirror duals of magnetic fluxes
• Unification of Type II Strings and T-duality
• Extended geometry and gauged maximal supergravity
• Non-geometric backgrounds, doubled geometry and generalised T-duality
• Asymmetric Orbifolds, Non-Geometric Fluxes and Non-Commutativity in Closed String Theory
• Current Algebras and Differential Geometry
• Torsion-free generalized connections and Heterotic Supergravity
• Algebroids, Heterotic Moduli Spaces and the Strominger System
• Leibniz algebroids, twistings and exceptional generalized geometry
• The Hitchin functionals and the topological B-model at one loop
• E11, generalised space-time and IIA string theory
• The gauge structure of Exceptional Field Theories and the tensor hierarchy
• Infinitesimal moduli for the Strominger system and Killing spinors in generalized geometry
• Generalised geometry, eleven dimensions and E11
• Special Hermitian metrics on compact solvmanifolds
• Generalized complex geometry and T-duality
• Killing-Yano tensors and multi-hermitian structures
• IIB Supergravity and the E6(6) covariant vector-tensor hierarchy
• A surgery for generalized complex structures on 4-manifolds
• Blow-up of generalized complex 4-manifolds
• N=1 vacua in Exceptional Generalized Geometry
• E11, generalised space-time and equations of motion in four dimensions
• Generalized complex geometry and the Poisson Sigma Model
• Supersymmetric Backgrounds, the Killing Superalgebra, and Generalised Special Holonomy
• Brackets
• Mirror Symmetry, Hitchin's Equations, And Langlands Duality
• Introductory Lectures on Extended Supergravities and Gaugings
• The Pontryagin Class for Pre-Courant Algebroids
• The generalized Kaehler geometry of N=(2,2) WZW-models
• Lie Algebroids and the Geometry of Off-shell BRST
• Leibniz algebroids, generalized Bismut connections and Einstein-Hilbert actions
• Gauging the Poisson sigma model
• Supersymmetric WZ-Poisson sigma model and twisted generalized complex geometry
• Twistorial construction of generalized Kaehler manifolds
• On the Geometry of Multi-Dirac Structures and Gerstenhaber Algebras
• Probing the String Winding Sector
• Generalising G${}_{\text{2}}$ geometry: involutivity, moment maps and moduli
• Star products on graded manifolds and ${\alpha }^{\prime }$-corrections to Courant algebroids from string theory
• Semichiral Fields on S^2 and Generalized Kähler Geometry
• Monge-Ampere equations and generalized complex geometry. The two-dimensional case
• T-duality of current algebras and their quantization
• Towards a world-sheet description of doubled geometry in string theory
• Index type invariants for twisted signature complexes and homotopy invariance
• Isotropic A-branes and the stability condition
• Topological T-duality for torus bundles with monodromy
• A Note on Supersymmetric Type II Solutions of Lifshitz Type
• Generalized CoKähler Geometry and an Application to Generalized Kähler Structures
• Invariant Generalized Complex Structures on Flag Manifolds
• Glanon groupoids
• Electric-Magnetic Dualities in Non-Abelian and Non-Commutative Gauge Theories
• A Curvature Flow Unifying Symplectic Curvature Flow And Pluriclosed Flow
• Pure Subspaces, Generalizing the Concept of Pure Spinors
• BRST symmetry of doubled membrane sigma-models
• The GIT aspect of generalized K$\ddot{a}$hler reduction. I
• Graded Generalized Geometry
• A Simple Model of Double Dynamics on Lie Groups
• Courant bracket twisted both by a 2-form $B$ and by a bi-vector $\theta$
• Moser Lemma in Generalized Complex Geometry
• Variation of Hodge structure for generalized complex manifolds
• From Hypercomplex to Holomorphic Symplectic Structures
• Higher curvature Bianchi identities, generalised geometry and $L_{\infty }$ algebras
• Toric generalized Kaehler structures. III
• Invariant generalized complex geometry on maximal flag manifolds and their moduli
• Complex Dirac structures: invariants and local structure
• Metric Reduction and Generalized Holomorphic Structures
• Causal transformation of Gödel-type space-times in conformal field theory
• Conformally Invariant Spinorial Equations in Six Dimensions
• Double Field Theory and Pseudo-Supersymmetry
• Dimensional Reductions of DFT and Mirror Symmetry for Calabi-Yau Three-folds and $K3\times T^2$
• Lagrangian formulation for electric charge in a magnetic monopole distribution
• Generalized Luttinger surgery and other cut-and-paste constructions in generalized complex geometry