Chern-Simons invariant in nLab

∞-Chern-Weil theory

∞-Chern-Simons theory

∞-Wess-Zumino-Witten theory

Ingredients

• quantum field theory

• circle n-bundle with connection

• ∞-Chern-Weil theory

  • ∞-Lie algebroid cohomology

  • Chern-Simons element

Definition

• level (Chern-Simons theory)

Examples

• For semisimple Lie algebra targets

  • Chern-Simons theory

• For discrete group targets

  • Dijkgraaf-Witten theory

• For discrete 2-group targets

  • Crane-Yetter model

• For Lie 2-algebra targets

  • BF-theory coupled to topological Yang-Mills theory

• For targets extending the super Poincare Lie algebra

  (such as the supergravity Lie 3-algebra, the supergravity Lie 6-algebra)

  • Chern-Simons gravity

  • Chern-Simons-supergravity

• 1-dimensional Chern-Simons theory

• for higher abelian targets

  • higher dimensional Chern-Simons theory

  • higher electric background charge coupling

• for symplectic Lie n-algebroid targets

  • AKSZ theory

    • Poisson sigma model

      • A-model, B-model

    • Courant sigma model

      • Chern-Simons theory

    • topological Yang-Mills theory

• for the L ∞L_\infty-structure on the BRST complex of the closed string:

  • string field theory

• higher dimensional Chern-Simons theory

  • 1d Chern-Simons theory

    • Wilson loop

  • 2d Chern-Simons theory

    • Poisson sigma-model,

    • A-model, B-model topological string

  • 3d Chern-Simons theory

    • Chern-Simons theory

    • Dijkgraaf-Witten theory

    • Courant sigma-model

    • 3d quantum gravity

  • 4d Chern-Simons theory

    • Yetter model

    • BF-theory

  • 5d Chern-Simons theory

  • 6d Chern-Simons theory

  • 7d Chern-Simons theory

    • topological AdS7/CFT6-sector

    • topological M-theory

  • 11d Chern-Simons theory

    • RR-field

• ∞-Chern-Simons theory

  • ∞-Dijkgraaf-Witten theory

  • string field theory

  • AKSZ sigma-model

functorial quantum field theory

Contents

• cobordism category

  • cobordism

  • extended cobordism

  • (∞,n)-category of cobordisms

  • Riemannian bordism category

• cobordism hypothesis

  • generalized tangle hypothesis

  • classification of TQFTs

• functorial field theory

  • unitary functorial field theory

  • extended functorial field theory

  • CFT

    • vertex operator algebra

  • TQFT

    • Reshetikhin-Turaev model / Chern-Simons theory

    • HQFT

    • TCFT

      • A-model, B-model, Gromov-Witten theory

      • homological mirror symmetry

• FQFT and cohomology

  • (1,1)-dimensional Euclidean field theories and K-theory

  • (2,1)-dimensional Euclidean field theory

  • geometric models for tmf

• holographic principle of higher category theory

  • holographic principle

  • AdS/CFT correspondence

  • quantization via the A-model

physics, mathematical physics, philosophy of physics

Surveys, textbooks and lecture notes

• (higher) category theory and physics

• geometry of physics

• books and reviews, physics resources

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theory (physics), model (physics)

experiment, measurement, computable physics

• mechanics

  • mass, charge, momentum, angular momentum, moment of inertia

  • dynamics on Lie groups

    • rigid body dynamics

• field (physics)

  • Lagrangian mechanics

    • configuration space, state

    • action functional, Lagrangian

    • covariant phase space, Euler-Lagrange equations

  • Hamiltonian mechanics

    • phase space

    • symplectic geometry

      • Poisson manifold

      • symplectic manifold

      • symplectic groupoid

    • multisymplectic geometry

      • n-symplectic manifold

  • spacetime

    • smooth Lorentzian manifold

    • special relativity

    • general relativity

    • gravity

      • supergravity, dilaton gravity

      • black hole

• Classical field theory

  • classical physics

    • classical mechanics
    • waves and optics
    • thermodynamics

• Quantum Mechanics

  • in terms of ∞-compact categories

  • quantum information

  • Hamiltonian operator

  • density matrix

  • Kochen-Specker theorem

  • Bell's theorem

  • Gleason's theorem

• Quantization

  • geometric quantization

  • deformation quantization

  • path integral quantization

  • semiclassical approximation

• Quantum Field Theory

  • Axiomatizations

    • algebraic QFT

      • Wightman axioms

      • Haag-Kastler axioms

        • operator algebra

        • local net

      • conformal net

      • Reeh-Schlieder theorem

      • Osterwalder-Schrader theorem

      • PCT theorem

      • Bisognano-Wichmann theorem

        • modular theory

      • spin-statistics theorem

        • boson, fermion

    • functorial QFT

      • cobordism

      • (∞,n)-category of cobordisms

      • cobordism hypothesis-theorem

      • extended topological quantum field theory

  • Tools

    • perturbative quantum field theory, vacuum

    • effective quantum field theory

    • renormalization

    • BV-BRST formalism

    • geometric ∞-function theory

  • particle physics

    • phenomenology

    • models

      • standard model of particle physics

      • fields and quanta

      • Grand Unified Theories, MSSM

    • scattering amplitude

      • on-shell recursion, KLT relations

  • Structural phenomena

    • universality class

    • quantum anomaly

      • Green-Schwarz mechanism

    • instanton

    • spontaneously broken symmetry

    • Kaluza-Klein mechanism

    • integrable systems

    • holonomic quantum fields

  • Types of quantum field thories

    • TQFT

      • 2d TQFT

      • Dijkgraaf-Witten theory

      • Chern-Simons theory

    • TCFT

      • A-model, B-model

      • homological mirror symmetry

    • QFT with defects

    • conformal field theory

    • (1,1)-dimensional Euclidean field theories and K-theory

    • (2,1)-dimensional Euclidean field theory and elliptic cohomology

    • CFT

      • WZW model

      • 6d (2,0)-supersymmetric QFT

    • gauge theory

      • field strength

      • gauge group, gauge transformation, gauge fixing

      • examples

        • electromagnetic field, QED

      • electric charge

      • magnetic charge

        • Yang-Mills field, QCD

      • Yang-Mills theory

      • spinors in Yang-Mills theory

      • topological Yang-Mills theory

        • Kalb-Ramond field
        • supergravity C-field
        • RR field
        • first-order formulation of gravity

      • general covariance

      • supergravity

      • D'Auria-Fre formulation of supergravity

      • gravity as a BF-theory

    • sigma-model

      • particle, relativistic particle, fundamental particle, spinning particle, superparticle

      • string, spinning string, superstring

      • membrane

      • AKSZ theory

• String Theory

  • string theory results applied elsewhere

• number theory and physics

  • Riemann hypothesis and physics