nLab
geometryofphysicscontents

  • About this page

    • Scope and perspective
    • Layers of exposition
    • The full story in a few formal words
  • Coordinate systems

    • Model Layer

      • The continuum real (world-)line
      • Cartesian spaces and smooth functions
      • The fundamental theorems about smooth functions
    • Semantic Layer

      • The algebraic theory of smooth algebras
      • The coverage of differentially good open covers
      • The slice category
    • Syntactic Layer

      • Judgments about types and terms
      • Natural deduction rules for product types
      • Natural deduction rules for dependent sum types
      • Dictionary: type theory / category theory
  • Smooth spaces

    • Model Layer

      • Plots of smooth spaces and their gluing
      • Homomorphisms of smooth spaces
      • Products and fiber products of smooth spaces
      • Smooth mapping spaces and smooth moduli spaces
      • The smooth moduli space of smooth functions
      • Outlook
    • Semantic Layer

      • Toposes

      • Subobjects

      • Slice toposes

      • Local, connected and cohesive toposes

      • The topos of smooth spaces

        • Connectedness
        • Locality
        • Cohesion
    • Syntactic Layer

      • Natural deduction rules for dependent product types
      • Internal logic of a toposThe type of propositions
      • Cohesive modality I: Sharp types
  • Differential forms

    • Model Layer

      • Differential forms on abstract coordinate systems
      • Differential forms on smooth spaces
    • Semantic Layer

      • Concrete smooth spaces
      • Smooth space of differential forms on a smooth space
    • Syntactic Layer

      • Images
  • Differentiation

    • Model Layer

      • Differentiation of smooth functions and differential forms

        • Differentiation on coordinate patches
        • Differentiation on smooth spaces
        • Example: The electromagnetic field strength
      • Variational calculus

        • Discrete points of a smooth space
        • Smooth functionals
        • Functional derivative / variational derivative
        • Euler-Lagrange equations
      • π’Ÿ\mathcal{D}-geometry

        • Infinitesimal smooth loci
        • de Rham space
        • Jet bundles
    • Semantic Layer

      • Synthetic differential geometry
      • Tangent bundle
      • Differential equations
      • Differential cohesion of the topos of smooth spaces
      • Differential cohesion
      • de Rham space
      • Jet bundle
      • Formally Γ©tale / formally unramified / formally smooth
    • Syntactic Layer

      • Differential homotopy type theory
  • Smooth homotopy types

    • Model Layer

      • Gauge transformations in electromagnetism
      • Groupoids
      • Smooth groupoids
      • Differential 1-forms are smooth incremental path measures
    • Semantic Layer

      • ∞-Toposes
      • Slice ∞-toposes
      • Local, ∞-connected and cohesive ∞-toposes
      • The ∞-topos of smooth ∞-groupoids
      • Local ∞-Connectedness
      • Locality
      • Cohesion
    • Syntactic Layer

      • Identity types
      • Homotopy type theory
      • Cohesive modality II: flat types
  • Groups

    • Model Layer

      • Groups
      • Dold-Kan correspondence
    • Semantic Layer

      • A ∞-types
      • ∞-Groups
    • Syntactic Layer

      • Pointed connected types
      • Identity types of connected types
  • Principal bundles

    • Model Layer

      • Connected groupoids
      • Universal principal bundle
      • Principal bundles
      • Weakly principal simplicial bundles
    • Semantic Layer

      • Principal ∞-bundles
    • Syntactic Layer

  • Actions, representations and associated bundles

    • Model Layer

      • Actions
      • Associated bundle
      • Representations up to coherent homotopy
    • Semantic Layer

      • ∞-Actions
      • Associated ∞-bundles
    • Syntactic Layer

      • The context of a pointed connected type: representation theory
      • Dependent product over a pointed connected type: invariants
      • Dependent sum over a pointed connected type: quotients
  • Manifolds

    • Model Layer

      • Smooth manifolds
      • Tangent bundle
    • Semantic Layer

      • Manifolds modeled on an object
    • Syntactic Layer

  • GG-Structure

    • Model Layer

      • Reduction of structure group
      • Vielbein, orthogonal structure
      • Almost complex structure
      • Almost Hermitean structure
      • Almost symplectic structure
      • Metaplectic structure
      • Metalinear structure
      • Generalized complex geometry
      • Type II geometry
      • Generalized Calabi-Yau structure
      • Exceptional generalized geometry
      • Spin structure, String structure, Fivebrane structure
    • Semantic Layer

    • Syntactic Layer

  • Riemannian geometry

    • Model Layer
    • Semantic Layer
    • Syntactic Layer
  • Integration

    • Model Layer

      • Integration
      • Integration of differential forms
      • Integration in ordinary differential cohomology
      • Lie integration
      • Transgression
      • Transgression of differential forms
      • Transgression of circle n-bundles with connection
      • Action functionals from transgression
    • Semantic Layer

    • Syntactic Layer

  • Flat connections

    • Model Layer
    • Semantic Layer
    • Syntactic Layer
  • de Rham Coefficients

    • Model Layer

      • Lie-algebra valued differential 1-forms
      • The de Rham complex
    • Semantic Layer

      • De Rham coefficient objects
      • Recovering smooth differential forms from cohesive de Rham coefficients
    • Syntactic Layer

  • Maurer-Cartan forms

    • Model Layer

      • Maurer-Cartan form on a Lie group

    Semantic Layer

    • Maurer-Cartan form on a cohesive ∞-group

    • Maurer-Cartan forms on smooth ∞-groups

    • Cohesive differentiation

    • Universal curvature characteristic forms

    • Syntactic Layer

  • Circle-principal connections

    • Model Layer

      • Circle-principal connections
      • Dirac charge quantization and the electromagnetic field
      • Deligne cohomology and Cheeger-Simons differential characters
      • Circle-principal 2-connection
      • Magnetic charge and the B-field
      • Circle-principal 3-connection
      • The 3d Chern-Simons action functional and the C-field
    • Semantic Layer

      • Smooth differential cohomology
      • Higher holonomy
    • Syntactic Layer

  • Characteristic classes

    • Model Layer
    • Semantic Layer
    • Syntactic Layer
  • Circle-principal n-connection

    • Model Layer
    • Semantic Layer
    • Syntactic Layer
  • Action functionals for Chern-Simons-type gauge theories

    • Model Layer
    • Semantic Layer
    • Syntactic Layer
  • Abelian Chern-Simons theory

    • Model Layer
    • Semantic Layer
    • Syntactic Layer
  • Principal connections

    • Model Layer

      • Yang-Mills theory
    • Semantic Layer

    • Syntactic Layer

  • Associated bundles

    • Model Layer

      • Associated vector bundle
      • Spin geometry
    • Semantic Layer

    • Syntactic Layer

  • Covariant derivative

  • Einstein-Yang-Mills theory

  • Symplectic geometry

  • Geometric quantization

  • Supergeometric coordinate systems

    • Model Layer
    • Semantic Layer
    • Syntactic Layer
  • References

    • General
    • Differential forms and parallel transport
    • Mathematical quantum field theory
    • Further

Created on October 20, 2012 at 15:22:09. See the history of this page for a list of all contributions to it.