Lectures on Generalized Complex Geometry for Physicists

Abstract

In these lectures we review Generalized Complex Geometry and discuss two main applications to string theory: the description of supersymmetric flux compactifications and the supersymmetric embedding of D-branes. We start by reviewing G-structures, and in particular SU(3)-structure and its torsion classes, before extending to Generalized Complex Geometry. We then discuss the supersymmetry conditions of type II supergravity in terms of differential conditions on pure spinors, and finally introduce generalized calibrations to describe D-branes. As examples we discuss in some detail AdS4 compactifications, which play a role as the geometric duals in the AdS4/CFT3-correspondence.

94 pages, 4 figures, 5 tables, lectures Sogang University August 2007 and Modave Summer School September 2008, v2: references added

References in corpus (146)

• de Sitter Vacua in String Theory
• N=6 superconformal Chern-Simons-matter theories, M2-branes and their gravity duals
• Hierarchies from Fluxes in String Compactifications
• Supergravity description of field theories on curved manifolds and a no go theorem
• CFT's From Calabi-Yau Four-folds
• Four-dimensional String Compactifications with D-Branes, Orientifolds and Fluxes
• Flux Compactification
• Fivebranes, Membranes and Non-Perturbative String Theory
• Flux compactifications in string theory: a comprehensive review
• Super-Poincare Covariant Quantization of the Superstring
• A Geometry for Non-Geometric String Backgrounds
• Super D-branes
• The Dirichlet Super-p-Branes in Ten-Dimensional Type IIA and IIB Supergravity
• RR Flux on Calabi-Yau and Partial Supersymmetry Breaking
• The effective action of N=1 Calabi-Yau orientifolds
• Nongeometric Flux Compactifications
• Generalised Geometry for M-Theory
• Non-Kaehler String Backgrounds and their Five Torsion Classes
• Doubled Geometry and T-Folds
• Superstrings with Intrinsic Torsion
• New Formulations of D=10 Supersymmetry and D8-O8 Domain Walls
• The Geometry of D=11 Killing Spinors
• Generalized structures of N=1 vacua
• D-Brane Actions with Local Kappa Symmetry
• The effective action of Type IIA Calabi-Yau orientifolds
• Gauge-Invariant and Gauge-Fixed D-Brane Actions
• Lectures on Special Lagrangian Submanifolds
• Mirror Symmetry in Generalized Calabi-Yau Compactifications
• M-theory, exceptional generalised geometry and superpotentials
• Reduction of Courant algebroids and generalized complex structures
• The Dirichlet Super-Three-Brane in Ten-Dimensional Type IIB Supergravity
• A scan for new N=1 vacua on twisted tori
• Geometric Constructions of Nongeometric String Theories
• Supersymmetric AdS(4) compactifications of IIA supergravity
• Solitons, Superpotentials and Calibrations
• T-duality, Generalized Geometry and Non-Geometric Backgrounds
• Lectures on Gauged Supergravity and Flux Compactifications
• Derived brackets
• Hitchin Functionals in N=2 Supergravity
• Type II Theories Compactified on Calabi-Yau Threefolds in the Presence of Background Fluxes
• Remarks on A-branes, Mirror Symmetry, and the Fukaya category
• The geometry of three-forms in six and seven dimensions
• Towards Classical de Sitter Solutions in String Theory
• SU(3) x SU(3) compactification and mirror duals of magnetic fluxes
• Supersymmetric AdS_5 Solutions of Type IIB Supergravity
• From ten to four and back again: how to generalize the geometry
• D-branes on general N=1 backgrounds: superpotentials and D-terms
• Generalized N=1 Orientifold Compactifications and the Hitchin functionals
• Supersymmetric D-branes and calibrations on general N=1 backgrounds
• Moduli spaces of Chern-Simons quiver gauge theories and AdS_4/CFT_3
• Physics of String Flux Compactifications
• Dirac actions for D-branes on backgrounds with fluxes
• Instantons, Poisson structures and generalized Kaehler geometry
• D-brane Wess--Zumino actions, T-duality and the cosmological constant
• Generalized Flux Vacua
• The effective theory of type IIA AdS4 compactifications on nilmanifolds and cosets
• Mirror symmetric SU(3)-structure manifolds with NS fluxes
• Supersymmetry and generalized calibrations
• General N = 1 Supersymmetric Flux Vacua of (Massive) Type IIA String Theory
• Lectures on D-branes, Gauge Theory and M(atrices)
• Supersymmetric sources, integrability and generalized-structure compactifications
• Generalized non-supersymmetric flux vacua
• General N = 1 Supersymmetric Fluxes in Massive Type IIA String Theory
• A simple class of N=3 gauge/gravity duals
• Supersymmetric Cycles in Exceptional Holonomy Manifolds and Calabi-Yau 4-Folds
• On Slow-roll Moduli Inflation in Massive IIA Supergravity with Metric Fluxes
• An N = 2 Supersymmetric Membrane Flow
• On the Cosmology of Type IIA Compactifications on SU(3)-structure Manifolds
• The gauge dual of Romans mass
• ABJM with Flavors and FQHE
• Effective action of (massive) IIA on manifolds with SU(3) structure
• Notes on toric Sasaki-Einstein seven-manifolds and AdS_4/CFT_3
• Stable D-branes, calibrations and generalized Calabi-Yau geometry
• Minimal Simple de Sitter Solutions
• Exploiting N=2 in consistent coset reductions of type IIA
• Universal de Sitter solutions at tree-level
• Type IIA AdS4 compactifications on cosets, interpolations and domain walls
• A note on the holography of Chern-Simons matter theories with flavour
• Generalized complex manifolds and supersymmetry
• Generalized Kahler manifolds and off-shell supersymmetry
• Moduli Stabilization and Cosmology of Type IIB on SU(2)-Structure Orientifolds
• Gauged Supergravities from Twisted Doubled Tori and Non-Geometric String Backgrounds
• New supersymmetric AdS4 type II vacua
• Deformations of calibrated D-branes in flux generalized complex manifolds
• Supersymmetry breaking branes on solvmanifolds and de Sitter vacua in string theory
• Towards Minkowski Vacua in Type II String Compactifications
• Reformulating Supersymmetry with a Generalized Dolbeault Operator
• AdS Calibrations
• On moduli and effective theory of N=1 warped flux compactifications
• New string vacua from twistor spaces
• The non-abelian D-brane effective action through order ${\alpha }^{\prime }{}^4$
• Effective actions and N=1 vacuum conditions from SU(3) x SU(3) compactifications
• AdS_4/CFT_3 duals from M2-branes at hypersurface singularities and their deformations
• Massive type IIA string theory cannot be strongly coupled
• Topological mirror symmetry with fluxes
• The non-abelian open superstring effective action through order ${\alpha }^{\prime }{}^3$
• U-dual fluxes and Generalized Geometry
• Perturbing gauge/gravity duals by a Romans mass
• Calibrations and Intersecting Branes
• E7(7) formulation of N=2 backgrounds
• Superconformal Chern-Simons Theories and the Squashed Seven Sphere
• Mirror Symmetry and Generalized Complex Manifolds
• Nearly Kaehler Reduction
• Generalized geometry and the Hodge decomposition
• On the supergravity formulation of mirror symmetry in generalized Calabi-Yau manifolds
• T-duality with H-flux: non-commutativity, T-folds and G x G structure
• Classes of AdS4 type IIA/IIB compactifications with SU(3)xSU(3) structure
• Gravity duals to deformed SYM theories and Generalized Complex Geometry
• D-branes on AdS flux compactifications
• N=2 solutions of massive type IIA and their Chern-Simons duals
• D-brane networks in flux vacua, generalized cycles and calibrations
• Coisotropic D8-branes and Model-building
• Lectures on Generalized Complex Geometry and Supersymmetry
• Nonabelian Phenomena on D-branes
• New supersymmetric flux vacua with intermediate SU(2) structure
• Towards reduction of type II theories on SU(3) structure manifolds
• Non-geometric String Backgrounds and Worldsheet Algebras
• DWSB in heterotic flux compactifications
• N=1,2 supersymmetric vacua of IIA supergravity and SU(2) structures
• Generalized Kaehler Potentials from Supergravity
• Heterotic String Theory with Background Fluxes
• AdS_5 Solutions of Type IIB Supergravity and Generalized Complex Geometry
• Flux backgrounds from Twists
• Mirror symmetry for topological sigma models with generalized Kahler geometry
• NS-NS fluxes in Hitchin's generalized geometry
• On Deformations of Generalized Complex Structures: the Generalized Calabi-Yau Case
• Domain walls from ten dimensions
• The Quantum Geometry of N=(2,2) Non-Linear Sigma-Models
• Reducing democratic type II supergravity on SU(3) x SU(3) structures
• The 11-dimensional Metric for AdS/CFT RG Flows with Common SU(3) Invariance
• The biHermitian topological sigma model
• On the geometry and the moduli space of beta-deformed quiver gauge theories
• Flux compactification on smooth, compact three-dimensional toric varieties
• No-scale supersymmetry breaking vacua and soft terms with torsion
• Geometry of D-branes for general N=(2,2) sigma models
• Linearizing Generalized Kahler Geometry
• Abelian and Non-abelian D-brane Effective Actions
• Generalized Calabi-Yau structures and mirror symmetry
• Type II compactifications on manifolds with SU(2) x SU(2) structure
• A landscape of non-supersymmetric AdS vacua on coset manifolds
• Type IIA orientifold compactification on SU(2)-structure manifolds
• Topological Sigma Models with H-Flux
• Potentials for Light Moduli in Supergravity and String Theory
• An N=2 worldsheet approach to D-branes in bihermitian geometries: II. The general case
• An N=2 worldsheet approach to D-branes in bihermitian geometries: I. Chiral and twisted chiral fields
• Coisotropic D-branes on AdS4 x CP3 and massive deformations

Cited by in corpus (82)

• Double Field Theory: A Pedagogical Review
• Nonperturbative aspects of ABJM theory
• De Sitter hunting in a classical landscape
• Covariant action for a string in doubled yet gauged spacetime
• Stringy Unification of Type IIA and IIB Supergravities under N=2 D=10 Supersymmetric Double Field Theory
• Comments on double field theory and diffeomorphisms
• The problematic backreaction of SUSY-breaking branes
• Non-geometric backgrounds in string theory
• β-supergravity: a ten-dimensional theory with non-geometric fluxes, and its geometric framework
• Algebroids, Heterotic Moduli Spaces and the Strominger System
• U-geometry : SL(5)
• The Emergence Proposal in Quantum Gravity and the Species Scale
• Duality Symmetric String and M-Theory
• The gauge structure of Exceptional Field Theories and the tensor hierarchy
• All homogeneous N=2 M-theory truncations with supersymmetric AdS4 vacua
• Refining the boundaries of the classical de Sitter landscape
• Scale-separated AdS${}_4$ vacua of IIA orientifolds and M-theory
• dS Supergravity from 10d
• The N=1 effective actions of D-branes in Type IIA and IIB orientifolds
• The Type IIA Flux Potential, 4-forms and Freed-Witten anomalies
• D-brane non-perturbative effects and geometric deformations
• Large Field Inflation from Axion Mixing
• A new twist on heterotic string compactifications
• Twisted Cohomotopy implies M-theory anomaly cancellation on 8-manifolds
• Heterotic string from a higher dimensional perspective
• Kähler moduli stabilization from ten dimensions
• Soft branes in supersymmetry-breaking backgrounds
• Dynamic SU(2) Structure from Seven-branes
• SL(5) duality from canonical M2-brane
• D-branes in Generalized Geometry and Dirac-Born-Infeld Action
• M5 algebra and SO(5,5) duality
• Type IIB flux vacua from G-theory II
• Systematics of Type IIA moduli stabilisation
• Electrified branes
• New supersymmetric vacua on solvmanifolds
• Heterotic domain wall solutions and SU(3) structure manifolds
• On de Sitter Spacetime and String Theory
• Charting the landscape of (anti-) de Sitter and Minkowski solutions of 10d supergravities
• IIB supergravity on manifolds with SU(4) structure and generalized geometry
• Type IIA Flux Vacua with Mobile D6-branes
• Twisted Cohomotopy implies M5-brane anomaly cancellation
• Supersymmetry, T-duality and Heterotic ${\alpha }^{\prime }$-corrections
• Cosmological Implications from O(D,D)
• Gaugino mass term for D-branes and Generalized Complex Geometry
• Generalized Geometry in AdS/CFT and Volume Minimization
• Five-Brane Superpotentials, Blow-Up Geometries and SU(3) Structure Manifolds
• Spin(7)-manifolds in compactifications to four dimensions
• Subsectors, Dynkin Diagrams and New Generalised Geometries
• U-gravity : $\mathbf{SL}(N)$
• Compact G2 holonomy spaces from SU(3) structures
• Revisiting toric SU(3) structures
• New Examples of Flux Vacua
• D-brane on Poisson manifold and Generalized Geometry
• Semichiral Fields on S^2 and Generalized Kähler Geometry
• Weak $G_2$-manifolds and scale separation in M-theory from type IIA backgrounds
• Gauged supergravities and their symmetry-breaking vacua in F-theory
• D-branes and non-Abelian T-duality
• Smearing and Unsmearing KKLT AdS Vacua
• Non-supersymmetric branes
• Holomorphic Couplings In Non-Perturbative String Compactifications
• Exploring the landscape of (anti-) de Sitter and Minkowski solutions: group manifolds, stability and scale separation
• Integrable deformation of ${\mathbb{CP}}^n$ and generalised Kaehler geometry
• Towards Geometric D6-Brane Model Building on non-Factorisable Toroidal ${\mathbb{Z}}_4$-Orbifolds
• M-theory on non-Kähler manifolds
• Superspaces for heterotic pure spinor string compactifications
• SL(2,R) covariant conditions for N=1 flux vacua
• Integrable Hopf twists, marginal deformations and generalised geometry
• Generalized Cohomologies and Supersymmetry
• BPS-like potential for compactifications of heterotic M-theory?
• Massless Spectra and Gauge Couplings at One-Loop on Non-Factorisable Toroidal Orientifolds
• Generalized complex geometry of pure backgrounds in ten and eleven dimensions
• On gauged linear sigma models with torsion
• Generalised Geometry and Flux Vacua
• Generalized Compactification in Heterotic String Theory
• Warped Vacuum Statistics
• Flux Backgrounds and Exceptional Generalised Geometry
• Double Field Theory and Pseudo-Supersymmetry
• Generalized geometry applied to 4d-supergravity
• Fundamental Landau Levels in a Strongly Coupled Plasma
• Geometry of Membrane Sigma Models
• Bubble instability of mIIA on ${\mathrm{AdS}}_4\times S^6$
• The many faces of brane-flux annihilation