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

    • $\mathcal{D}$-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 étale / 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

• $G$-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