superalgebra and (synthetic ) supergeometry
Formalism
Definition
Spacetime configurations
Properties
Spacetimes
black hole spacetimes | vanishing angular momentum | positive angular momentum |
---|---|---|
vanishing charge | Schwarzschild spacetime | Kerr spacetime |
positive charge | Reissner-Nordstrom spacetime | Kerr-Newman spacetime |
Quantum theory
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 -dimensional Minkowski space may be understood as the quotient of the Poincare group by the spin group cover of Lorentz group – the translation group.
Analogously, the for each real irreducible spin representation the -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 by its CE-algebra “of left-invariant 1-forms” on its group manifold.
Let and let be a real spin representation of . See at Majorana representation for details.
The Chevalley-Eilenberg algebra is generated on
elements and of degree
and elements of degree
where is a spacetime index, and where is an index ranging over a basis of the chosen Majorana spinor representation .
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 (def.).
This defines the super Poincaré super Lie algebra. After discarding the terms involving 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. are identified with left invariant 1-forms on the super-translation group (= super Minkowski spacetime) as follows.
Let be a real spin representation and let be the canonical coordinates on the supermanifold underlying the super-Minkowski super translation group. Then the canonical super vielbein is the -valued super differential form with components
.
.
Notice that this then gives the above formula for the differential of the super-vielbein in def. as
The term is sometimes called the supertorsion of the super-vielbein , because the defining equation
may be read as saying that 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 which have “all indices contracted”. See also at torsion constraints in supergravity.
Notably we have
This remaining operation “” of the differential acting on Loretz scalars is sometimes denoted “”, 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 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.
We spell out the super-translation super Lie group-structure on the supermanifold underlying super Minkowski spacetime, hence equivalently of the quotient super Lie group of the super Poincaré group (the “supersymmetry” group) by its Lorentzian spin-subgroup:
Here
is the spatial dimension (a natural number),
is a real spin representation equipped with a linear map
which is symmetric and -equivariant.
First, the super-Minkowski super Lie algebra structure on the super vector space
is defined, dually, by the Chevalley-Eilenberg dgc-superalgebra with generators of bidegree
generator | bidegree |
---|---|
for indexing a linear basis of and indexing a linear basis of by the differential equations
The first differential is the linear dual of the archetypical super Lie bracket in the supersymmetry super Lie algebra which takes two odd elements to a spatial translation. The second differential is the linear dual of the fact that in the absence of rotational generators, no Lie bracket in the supersymmetry alegbra results in a non-vanishing odd element.
Next we regard not just as a super vector space but as a Cartesian supermanifold. As such it has canonical coordinate functions
generator | bidegree |
---|---|
On this supermanifold, consider the super coframe field
(where on the left we have the tangent bundle and on the right its typical fiber super vector space) given by
It is clear that this is a coframe field in that for all it restricts to an isomorphism
and the peculiar second summand in the first line is chosen such that its de Rham differential has the same form as the differential in the Chevalley-Eilenberg algebra (2).
(Incidentally, a frame field linear dual to the coframe field (3) is
which are the operators often stated right away in introductory texts on supersymmetry.)
This fact, that the Maurer-Cartan equations of a coframe field (3) coincide with the defining equations (2) of the Chevalley-Eilenberg algebra of a Lie algebra of course characterizes the left invariant 1-forms on a Lie group, and hence what remains to be done now is to construct a super Lie group-structure on the supermanifold with respect to which the coframe (3) is left invariant 1-form.
Recalling (from here) that a morphism of supermanifolds is dually given by a reverse algebra homomorphism between their function algebras, which in the present case are freely generated by the above coordinate functions, we denote the canonical coordinates on the Cartesian product by for the first factor and for the second, and declare a group product operation as follows:
(cf. CAIP99, (2.1) & (2.6))
Here the choice of notation for the coordinates on the left is adapted to thinking of this group operation equivalently as the left multiplication action of the group on itself, which makes the following computation nicely transparent.
Indeed, the induced left action of the super-group on its odd tangent bundle
is dually given by
and left-invariance of the coframe (2) means that it is fixed by this operation (so the differential in the following computation is just that of the second factor, hence acting on unprimed coordinates only):
This shows that if (4) is the group product of a group object in SuperManifolds then the corresponding super Lie algebra is the super-Minkowski super translation Lie algebra and hence that this group object is the desired super-Minkowski super Lie group.
So, defining the remaining group object-operations as follows:
we conclude by checking the group object-axioms:
For associativity we need to check that the following diagram commutes:
and indeed it does — the term vanishes because the anti-commute among themselves, while the pairing (1) is symmetric:
For unitality we need to check that the following diagram commutes:
and indeed it does:
And finally, for invertibility we need to check that the following diagram commutes:
and indeed it does:
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--branes propagating on super-Minkowski space (FSS 13).
The corresponding -extensions are extended superspacetime.
Regarded as a super Lie algebra, super Minkowski spacetime 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 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 .
The , 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 September 3, 2024 at 19:00:44. See the history of this page for a list of all contributions to it.