nLab
extended super Minkowski spacetime
Contents
Context
Physics
∞ \infty -Chern-Weil theory
Super-Geometry
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 p p -branes .
Notably the supergravity Lie 3-algebra is the extended Minkowski spacetime on which superspacetimes of 11-dimensional supergravity are modeled.
geometric contextgauge group stabilizer subgroup local model space local geometry global geometry differential cohomology first order formulation of gravity differential geometry Lie group /algebraic group G G subgroup (monomorphism ) H ↪ G H \hookrightarrow G quotient (“coset space ”) G / H G/H Klein geometry Cartan geometry Cartan connection
examples Euclidean group Iso ( d ) Iso(d) rotation group O ( d ) O(d) Cartesian space ℝ d \mathbb{R}^d Euclidean geometry Riemannian geometry affine connection Euclidean 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 geometry pseudo-Riemannian geometry spin connection Einstein 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 d AdS^d AdS gravity
de Sitter group O ( d , 1 ) O(d,1) O ( d − 1 , 1 ) O(d-1,1) de Sitter spacetime dS d dS^d deSitter gravity
linear algebraic group parabolic subgroup /Borel subgroup flag variety parabolic geometry
conformal group O ( d , t + 1 ) O(d,t+1) conformal parabolic subgroup Möbius space S d , t S^{d,t} conformal geometry conformal connection conformal gravity
supergeometry super Lie group G G subgroup (monomorphism ) H ↪ G H \hookrightarrow G quotient (“coset space ”) G / H G/H super Klein geometry super Cartan geometry Cartan superconnection
examples super Poincaré group spin group super Minkowski spacetime ℝ d − 1 , 1 | N \mathbb{R}^{d-1,1\vert N} Lorentzian supergeometry supergeometry superconnection supergravity
super anti de Sitter group super anti de Sitter spacetime
higher differential geometry smooth 2-group G G 2-monomorphism H → G H \to G homotopy quotient G / / H G//H Klein 2-geometry Cartan 2-geometry
cohesive ∞-group ∞-monomorphism (i.e. any homomorphism ) H → G H \to G homotopy quotient G / / H G//H of ∞-action higher Klein geometry higher Cartan geometry higher Cartan connection
examples extended super Minkowski spacetime extended supergeometry higher 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
C. Chryssomalakos, José de Azcárraga , José M. Izquierdo , C. Pérez Bueno, The geometry of branes and extended superspaces , Nucl. Phys. B 567 (2000) 293-330 [arXiv:hep-th/9904137 , doi:10.1016/S0550-3213(99)00512-X ]
Makoto Sakaguchi, IIB-Branes and New Spacetime Superalgebras , JHEP 0004 (2000) 019 (arXiv:hep-th/9909143 )
José de Azcárraga , J. M. Izquierdo, Superalgebra cohomology, the geometry of extended superspaces and superbranes , Nucl.Phys.B567:293-330,2000 (arXiv:hep-th/0105125 )
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.
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