Lectures on Special Lagrangian Submanifolds

Abstract

These notes consist of a study of special Lagrangian submanifolds of Calabi-Yau manifolds and their moduli spaces. The particular case of three dimensions, important in string theory, allows us to introduce the notion of gerbes. These offer an appropriate language for describing many significant features of the Strominger-Yau-Zaslow approach to mirror symmetry.

Lectures given at the ICTP School on Differential Geometry April 1999

References in corpus (2)

• Special Lagrangian Fibrations I: Topology
• Special Lagrangian Fibrations II: Geometry

Cited by in corpus (124)

• The effective action of Type IIA Calabi-Yau orientifolds
• M-theory, exceptional generalised geometry and superpotentials
• Mirror symmetry, Langlands duality, and the Hitchin system
• WZW branes and gerbes
• Cluster decomposition, T-duality, and gerby CFT's
• Lectures on Generalized Complex Geometry for Physicists
• Twisted differential String and Fivebrane structures
• RR photons
• Higher Gauge Theory: 2-Connections on 2-Bundles
• Worldsheet approaches to D-branes on supersymmetric cycles
• Classical Chern-Simons theory, Part 1
• The effective action of type II Calabi-Yau orientifolds
• The Geometry of Bundle Gerbes
• Conformal Field Theories in Six-Dimensional Twistor Space
• Dirac Charge Quantization and Generalized Differential Cohomology
• The N=1 effective actions of D-branes in Type IIA and IIB orientifolds
• Abelian and non-Abelian branes in WZW models and gerbes
• Lectures on D-branes and Sheaves
• Non Abelian Differentiable Gerbes
• Special Lagrangian cones with higher genus links
• BRST, anti-BRST and gerbes
• Gerbes, M5-Brane Anomalies and E_8 Gauge Theory
• Supergravity as Generalised Geometry II: $E_{d(d)}\times {\mathbb{R}}^{+}$ and M theory
• Holonomy and parallel transport for Abelian gerbes
• Basic gerbe over non simply connected compact groups
• The spectral curve of the Eynard-Orantin recursion via the Laplace transform
• Generalized holomorphic bundles and the B-field action
• Collapsing D-Branes in Calabi-Yau Moduli Space: I
• Higher-Degree Analogs of the Determinant Line Bundle
• Generalized Kahler geometry
• The topology of T-duality for T^n-bundles
• Twisted K-theory and loop groups
• Gerbes on complex reductive Lie groups
• Global topology of the Hitchin system
• Discrete Torsion and Gerbes II
• Finite deformations from a heterotic superpotential: holomorphic Chern--Simons and an $L_{\infty }$ algebra
• Compactification, Geometry and Duality: N=2
• Abelian fibred holomorphic symplectic manifolds
• On D-brane moduli stabilisation
• On Twistors and Conformal Field Theories from Six Dimensions
• New N=1 dualities from orientifold transitions - Part II: String Theory
• Analogues of Discrete Torsion for the M-Theory Three-Form
• Differential Geometry of Gerbes
• U(1) mixing and D-brane linear equivalence
• G-gerbes, principal 2-group bundles and characteristic classes
• Differential cohomology and locally covariant quantum field theory
• Lattice Gerbe Theory
• Topological mirror symmetry for parabolic Higgs bundles
• Gaugino condensation in an improved heterotic M-theory
• Symmetry-deforming interactions of chiral p-forms
• Fivebrane Gravitational Anomalies
• General aspects of heterotic string compactifications on stacks and gerbes
• Modular curves, the Tate-Shafarevich group and Gopakumar-Vafa invariants with discrete charges
• Nonlinear PDEs for Fredholm determinants arising from string equations
• Discrete Torsion and Symmetric Products
• Mirror symmetry without corrections
• BIonic membranes and AdS instabilities
• Cycles, submanifolds, and structures on normal bundles
• TQFT's and gerbes
• Backreacted T-folds and non-geometric regions in configuration space
• D-branes and Azumaya noncommutative geometry: From Polchinski to Grothendieck
• Classifying orientifolds by flat n-gerbes
• A Cubical Set Approach to 2-Bundles with Connection and Wilson Surfaces
• G2 Hitchin functionals at one loop
• Topological Sigma Models with H-Flux
• Multiplicative vector fields on bundle gerbes
• Aspects of M-Theory Brane Interactions and String Theory Symmetries
• 2-Gerbes bound by complexes of gr-stacks, and cohomology
• Central extensions of gauge transformation groups of higher abelian gerbes
• Equivariant Dixmier-Douady Classes
• Index theory, eta forms, and Deligne cohomology
• Determinant bundles, boundaries, and surgery
• Non-Abelian Gerbes from Strings on a Branched Space-Time
• Charting the space of ground states with tensor networks
• Geometric Aspects of Mirror Symmetry (with SYZ for Rigid CY manifolds)
• Kernel algebras and generalized Fourier-Mukai transforms
• Nontrivial Azumaya noncommutative schemes, morphisms therefrom, and their extension by the sheaf of algebras of differential operators: D-branes in a $B$-field background à la Polchinski-Grothendieck Ansatz
• Deformation Expression for Elements of Algebras (II) --(Weyl algebra of 2m-generators)--
• p-Gerbes and Extended Objects in String Theory
• Topological G${}_2$ and Spin(7) strings at 1-loop from double complexes
• Expressions of algebra elements and transcendental noncommutative calculus
• Gerbes on orbifolds and exotic smooth R^4
• D-branes of A-type, their deformations, and Morse cobordism of A-branes on Calabi-Yau 3-folds under a split attractor flow: Donaldson/Alexander-Hilden-Lozano-Montesinos-Thurston/Hurwitz/Denef-Joyce meeting Polchinski-Grothendieck
• SYZ duality for parabolic Higgs moduli spaces
• Gerbes, covariant derivatives, p-form lattice gauge theory, and the Yang-Baxter equation
• Basic equivariant gerbes on non-simply connected compact simple Lie groups
• Bundle gerbes: stable isomorphism and local theory
• Towards an extended/higher correspondence -- Generalised geometry, bundle gerbes and global Double Field Theory
• A note on the holonomy of connections in twisted bundles
• Differential Cocycles and Dixmier-Douady Bundles
• Generalized almost paracontact structures
• A global Torelli theorem for rigid hyperholomorphic sheaves
• Bundle gerbes on supermanifolds
• Cyclic cocycles on twisted convolution algebras
• Quasi-quantum groups from Kalb-Ramond fields and magnetic amplitudes for strings on orbifolds
• Magnetic Backgrounds from Generalised Complex Manifolds
• Geometry of the N=2 supersymmetric sigma model with Euclidean worldsheet
• k-Gerbes, Line Bundles and Anomalies
• From Sparks to Grundles--Differential Characters
• Topological T-duality and Kaluza-Klein Monopoles
• Special Holonomy and Two-Dimensional Supersymmetric Sigma-Models
• Geometric and topological aspects of Type IIB D-branes
• Large Radius Limit and SYZ Fibrations of Hyper-Kahler Manifolds
• Weak mirror symmetry of Lie algebras
• Real Forms of Holomorphic Hamiltonian Systems
• Analytic Pontryagin Duality
• Eigenvalue crossings in Floquet topological systems
• Comparing two approaches to the K-theory classification of D-branes
• Degenerations and Lagrangian fibrations of Calabi-Yau manifolds
• Gravitational monopoles, anomalies and M$E_8$ bundles
• Spectral multiplicity and odd K-theory
• Transgression of the index gerbe
• Gerbes on $G_2$ Manifolds
• Supersymmetric Gauge Theories from String Theory
• Remarks on mirror symmetry of Donaldson-Thomas theory for Calabi-Yau 4-folds
• Abelian gerbes, generalized geometries and foliations of small exotic R^4
• Gerbes in Geometry, Field Theory, and Quantisation
• Heterotic String Data and Theta Functions
• Flux Backgrounds and Exceptional Generalised Geometry
• An Introduction to Gerbes on Orbifolds
• Closed non-abelian strings
• Rigidified torsor cocycles, hypercoverings and bundle gerbes
• NS Fivebrane and Tachyon Condensation
• Homotopy quantum field theory and the index gerbe