nLab Supersymmetry for mathematicians (changes)

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

spin geometry, string geometry, fivebrane geometry …

Ingredients

• supergeometry

• higher differential geometry

Spin geometry

spin geometry

• spin group

  pin group

  semi-spin group

  central product spin group

  spin^c group

• spin representation,

  • spinor, fermion

  • charge conjugation matrix, Fierz identity

  • real spin representation

  • Dirac conjugate

  • Dirac spinor, Weyl spinor, Majorana spinor

• spin structure, spin^c structure

• spinor bundle

• Dirac operator

  • index theory, K-theory

• Dirac equation

• Dirac field

rotation groups in low dimensions:

see also

• Spin(5).Spin(3)

• finite rotation groups

• ADE-classification

String geometry

string geometry

• string 2-group, string^c 2-group

• string structure, string^c structure

Fivebrane geometry

• fivebrane 6-group

• fivebrane structure

Ninebrane geometry

• ninebrane 10-group