super spacetime (Rev #4, changes) in nLab

gravity, supergravity

Formalism

• Poincaré Lie algebra

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

• differential geometry

• (super-)Cartan geometry

  • pseudo-Riemannian manifold

  • spacetime

Definition

• Einstein-Hilbert action, Einstein's equations, general relativity

  • first-order formulation of gravity, D'Auria-Fre formulation of supergravity

  • gravity as a BF-theory, Plebanski formulation of gravity, teleparallel gravity

Spacetime configurations

• spacetime

• Minkowski spacetime

• black hole spacetime

  • Schwarzschild spacetime

  • Kerr spacetime

Properties

• Kaluza-Klein mechanism

• Bekenstein-Hawking entropy

• generalized second law of thermodynamics

• cosmic censorship hypothesis

• weak gravity conjecture

Spacetimes

• standard model of cosmology

• Minkowski spacetime

• asymptotically flat spacetime

• gravitational wave

• anti de Sitter spacetime

• FRW spacetime

• Taub-NUT spacetime

• Einstein manifold

• pp-wave spacetime

• KK-monopole

• Malament–Hogarth spacetime

• super spacetime

  • super Minkowski spacetime

• black brane

Quantum theory

• quantum gravity

  • graviton, gravitino

  • string theory

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