superalgebra and (synthetic ) supergeometry

Background

• algebra

• geometry

• graded object

Introductions

• geometry of physics -- superalgebra

• geometry of physics -- supergeometry

Superalgebra

• super commutative monoid

• super abelian group

• super ring

  • supercommutative ring

  • exterior ring

  • Clifford ring

• super module

• super vector space, SVect

• super algebra

  • supercommutative algebra

  • exterior algebra

  • Clifford algebra

• superdeterminant

• super Lie algebra

  • super Poincare Lie algebra

• chain complex of super vector spaces (model structure)

• differential graded-commutative superalgebra (model structure)

• super L-infinity algebra

Supergeometry

• superpoint

• super Cartesian space

• supermanifold, SDiff

  • NQ-supermanifold

  • super vector bundle

  • complex supermanifold

  • Euclidean supermanifold

• super spacetime

  • super Minkowski spacetime

• integration over supermanifolds

  • Berezin integral

• super Lie group

  • super translation group

  • super Euclidean group

• super ∞-groupoid

• super formal smooth ∞-groupoid

• super line 2-bundle

Supersymmetry

supersymmetry

• division algebra and supersymmetry

• super Poincare Lie algebra

• supermultiplet

• BPS state

• M-theory super Lie algebra, type II super Lie algebra

• supergravity Lie 3-algebra, supergravity Lie 6-algebra

Supersymmetric field theory

• superfield

• supersymmetric quantum mechanics

  • superparticle

  • adinkra

• super Yang-Mills theory

• supergravity

  • gauged supergravity

• superstring theory

  • type II string theory

  • heterotic string theory

Applications

• geometric model for elliptic cohomology

physics, mathematical physics, philosophy of physics

Surveys, textbooks and lecture notes

• (higher) category theory and physics

• geometry of physics

• books and reviews, physics resources

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