superalgebra and (synthetic ) supergeometry
black hole spacetimes | vanishing angular momentum | positive angular momentum |
---|---|---|
vanishing charge | Schwarzschild spacetime | Kerr spacetime |
positive charge | Reissner-Nordstrom spacetime | Kerr-Newman spacetime |
Super-Minkowski spacetime is a super spacetime which is an analog in supergeometry of ordinary Minkowski spacetime. It is a super Cartesian space whose odd coordinates form a real spin representation.
Ordinary $(d+1)$-dimensional Minkowski space may be understood as the quotient $Iso(\mathbb{R}^{d-1,1})/(Spin(d-1,1))$ of the Poincare group by the spin group cover of Lorentz group – the translation group.
Analogously, the for each real irreducible spin representation $N$ the $dim(N)$-extended supermanifold Minkowski superspace or super Minkowski space is the quotient of supergroups of the super Poincaré group by the corresponding spin group (a super Klein geometry).
The super-translation group. See there for more details.
Alternatively, regarded as a super Lie algebra this is the quotient of the super Poincaré Lie algebra by the relevant Lorentz Lie algebra.
We briefly review some basics of the canonical coordinates and the super Lie algebra cohomology of the super Poincaré Lie algebra and super Minkowski space (see also at super Cartesian space and at signs in supergeometry).
By the general discussion at Chevalley-Eilenberg algebra, we may characterize the super Poincaré Lie algebra $\mathfrak{Iso}(\mathbb{R}^{d-1,1|N})$ by its CE-algebra $CE(\mathfrak{Iso}(\mathbb{R}^{d-1,1|N}))$ “of left-invariant 1-forms” on its group manifold.
Let $d \in \mathbb{N}$ and let $N$ be a real spin representation of $Spin(d-1,1)$. See at Majorana representation for details.
The Chevalley-Eilenberg algebra $CE(\mathfrak{Iso}(\mathbb{R}^{d-1,1|N}))$ is generated on
elements $\{e^a\}$ and $\{\omega^{ a b}\}$ of degree $(1,even)$
and elements $\{\psi^\alpha\}$ of degree $(1,odd)$
where $a \in \{0,1, \cdots, d-1\}$ is a spacetime index, and where $\alpha$ is an index ranging over a basis of the chosen Majorana spinor representation $N$.
The CE-differential defined as follows
and
(which is the differential for the semidirect product of the Poincaré Lie algebra acting on the given Majorana spinor representation)
and
where on the right we have the spinor-to-vector pairing in $N$ (def.).
This defines the super Poincaré super Lie algebra. After discarding the terms involving $\omega$ this becomes the CE algebra of the super translation algebra underlying super Minkowski spacetime.
In this way the super-Poincaré Lie algebra and its extensions is usefully discussed for instance in (D’Auria-Fré 82) and in (Azcárraga-Townsend 89, CAIB 99). In much of the literature instead the following equivalent notation is popular, which more explicitly involves the coordinates on super Minkowski space.
The abstract generators in def. 1 are identified with left invariant 1-forms on the super-translation group (= super Minkowski spacetime) as follows.
Let $N$ be a real spin representation and let $(x^a, \theta^\alpha)$ be the canonical coordinates on the supermanifold $\mathbb{R}^{d-1,1\vert N}$ underlying the super-Minkowski super translation group. Then the canonical super vielbein is the $\mathbb{R}^{d-1,1\vert N}$-valued super differential form with components
$\psi^\alpha \coloneqq \mathbf{d} \theta^\alpha$.
$e^a \coloneqq \mathbf{d} x^a + \overline{\theta} \Gamma^a \mathbf{d} \theta$.
Notice that this then gives the above formula for the differential of the super-vielbein in def. 1 as
The term $\frac{i}{2}\bar \psi \Gamma^a \psi$ is sometimes called the supertorsion of the super-vielbein $e$, because the defining equation
may be read as saying that $e$ is torsion-free except for that term. Notice that this term is the only one that appears when the differential is applied to “Lorentz scalars”, hence to object in $CE(\mathfrak{siso})$ which have “all indices contracted”. See also at torsion constraints in supergravity.
Notably we have
This remaining operation “$e \mapsto \Psi^2$” of the differential acting on Loretz scalars is sometimes denoted “$t_0$”, e.g. in (Bossard-Howe-Stelle 09, equation (8)).
This relation is what govers all of the exceptional super Lie algebra cocycles that appear as WZW terms for the Green-Schwarz action below: for some combinations of $(D,p)$ a Fierz identity implies that the term
vanishes identically, and hence in these dimensions the term
is a cocycle. See also the brane scan table below.
As opposed to ordinary Minkowski space, the de Rham cohomology of left invariant forms of super-Minkowski space contains nontrivial exceptional cocycles (the brane scan). These serve as the WZW terms for the Green-Schwarz action functional (see there for more) of super-$p$-branes propagating on super-Minkowski space (FSS 13).
The corresponding $L_\infty$-extensions are extended superspacetime.
Regarded as a super Lie algebra, super Minkowski spacetime $\mathbb{R}^{d-1,1|N}$ has the single nontrivial super-Lie bracket given by the spinor bilinear pairing
discussed in detail at spin representation.
Notice that this means that if one regards the superpoint $\mathbb{R}^{0|dim(N)}$ as an abelian super Lie algebra?, then super Minkowski spacetime is the Lie algebra extension of that by this bilinear pairing regarded as a super-Lie algebra cocycle with coefficients in $\mathbb{R}^{d}$.
The $d = 4$, $N =2$ super Minkowski spacetime was originally introduced in
Abdus Salam J.A. Strathdee, Supergauge Transformations, Nucl.Phys. B76 (1974) 477-482 (spire)
Abdus Salam J.A. Strathdee, Physical Review D11, 1521-1535 (1975)
see at “superspace in physics”.
Further discussion includes:
Super spacetimes and super Poincaré-group (pdf)
Daniel Freed, lecture 6 of Classical field theory and Supersymmetry, IAS/Park City Mathematics Series Volume 11 (2001) (pdf)
Daniel Freed, Lecture 4 of Five lectures on supersymmetry
Veeravalli Varadarajan, section 7 of Supersymmetry for mathematicians: An introduction
Leonardo Castellani, Riccardo D'Auria, Pietro Fre, page 370, part II section II.3.3 ofSupergravity and Superstrings - A Geometric Perspective_
Discussion of how super L-infinity algebra extensions of super Minkowski spacetime yield all the brane scan of string theory/M-theory is in
Last revised on April 23, 2018 at 08:16:48. See the history of this page for a list of all contributions to it.