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nLab extended super Minkowski spacetime

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Contents

Context

Physics

physics, mathematical physics, philosophy of physics

Surveys, textbooks and lecture notes

theory (physics), model (physics)

experiment, measurement, computable physics

∞\infty-Chern-Weil theory

Super-Geometry

superalgebra and (synthetic ) supergeometry

Background

Introductions

Superalgebra

Supergeometry

Supersymmetry

supersymmetry

Supersymmetric field theory

Applications

Contents

Idea

As opposed to ordinary Minkowski spacetime, super Minkowski spacetime has, when regarded as the super translation Lie algebra, exceptional cocycles. The super L-infinity algebra extensions classified by these serve as extended super-spacetimes which are target spaces for super pp-branes.

Notably the supergravity Lie 3-algebra is the extended Minkowski spacetime on which superspacetimes of 11-dimensional supergravity are modeled.

geometric contextgauge groupstabilizer subgrouplocal model spacelocal geometryglobal geometrydifferential cohomologyfirst order formulation of gravity
differential geometryLie group/algebraic group GGsubgroup (monomorphism) H↪GH \hookrightarrow Gquotient (“coset space”) G/HG/HKlein geometryCartan geometryCartan connection
examplesEuclidean group Iso(d)Iso(d)rotation group O(d)O(d)Cartesian space ℝ d\mathbb{R}^dEuclidean geometryRiemannian geometryaffine connectionEuclidean gravity
Poincaré group Iso(d−1,1)Iso(d-1,1)Lorentz group O(d−1,1)O(d-1,1)Minkowski spacetime ℝ d−1,1\mathbb{R}^{d-1,1}Lorentzian geometrypseudo-Riemannian geometryspin connectionEinstein gravity
anti de Sitter group O(d−1,2)O(d-1,2)O(d−1,1)O(d-1,1)anti de Sitter spacetime AdS dAdS^dAdS gravity
de Sitter group O(d,1)O(d,1)O(d−1,1)O(d-1,1)de Sitter spacetime dS ddS^ddeSitter gravity
linear algebraic groupparabolic subgroup/Borel subgroupflag varietyparabolic geometry
conformal group O(d,t+1)O(d,t+1)conformal parabolic subgroupMöbius space S d,tS^{d,t}conformal geometryconformal connectionconformal gravity
supergeometrysuper Lie group GGsubgroup (monomorphism) H↪GH \hookrightarrow Gquotient (“coset space”) G/HG/Hsuper Klein geometrysuper Cartan geometryCartan superconnection
examplessuper Poincaré groupspin groupsuper Minkowski spacetime ℝ d−1,1|N\mathbb{R}^{d-1,1\vert N}Lorentzian supergeometrysupergeometrysuperconnectionsupergravity
super anti de Sitter groupsuper anti de Sitter spacetime
higher differential geometrysmooth 2-group GG2-monomorphism H→GH \to Ghomotopy quotient G//HG//HKlein 2-geometryCartan 2-geometry
cohesive ∞-group∞-monomorphism (i.e. any homomorphism) H→GH \to Ghomotopy quotient G//HG//H of ∞-actionhigher Klein geometryhigher Cartan geometryhigher Cartan connection
examplesextended super Minkowski spacetimeextended supergeometryhigher supergravity: type II, heterotic, 11d

References

Discussion in the language of the D'Auria-Fre formulation of supergravity (“FDA”s) and the brane scan is in

and particularly with emphasis of the extended super Minkowski spacetimes as targets for the Green-Schwarz super-p brane sigma models with gauge fields on the worldvolume (tensor multiplets) in

Discussion that makes the super L-infinity algebra homotopy theory underlying this manifest is in

Last revised on June 19, 2023 at 14:41:26. See the history of this page for a list of all contributions to it.

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