superalgebra and (synthetic ) supergeometry
What is called $\kappa$-symmetry in string theory is a certain fermionic symmetry of Green-Schwarz action functionals for super p-branes whose effect is to gauge away half of the spinorial sigma-model fields.
In a completely super-covariant formulation of the Green-Schwarz action functionals, this $\kappa$-symmetry is simply the odd-graded part of the super-worldvolume super-diffeomorphism symmetry of the sigma-model (Sorokin-Tkach-Volkov 89, review includes Sorokin 99, section 4.3, Howe-Sezgin 04, section 4.3): If
$X$ denotes a superspacetime locally modeled on super-Minkowski spacetime $\mathbb{R}^{d-1,1\vert \mathbf{N}}$
$\Sigma$ denotes a super-worldvolume of a super p-brane locally modeled on super-Minkowski spacetime $\mathbb{R}^{p,1\vert \mathbf{N}/2}$
so that a sigma-model field configuration for a super p-brane of shape $\Sigma$ to popagate in $X$ is a morphism of supermanifolds of the form
then:
the postcomposition action of spacetime super-isometries $X \stackrel{\simeq}{\longrightarrow} X$ is in even degree the action of spacetime isometries and in odd degree the action of spacetime supersymmetry on the sigma-model fields;
the precomposition action of worldvolume super-diffeomorphism $\Sigma \stackrel{\simeq}{\to} \Sigma$ is in even degree the action of bosonic worldvolume diffeomorphism and in odd degree the action of $\kappa$-symmetry:
Notice here the assumption that the number of odd directions on the worldvolume is half that of the target spacetime. This is the default assumption for fundamental super p-branes, and it directly reflects the statement that the corrresponding black brane solutions are $1/2$ supergravity BPS states.
For example, consider the embedding
of 2+1d Minkowski spacetime, thought of as the worldvolume of a membrane, into 11d Minkowski spacetime, linearly along the coordinate axis. Any such embedding breaks the isometry group of $\mathbb{R}^{10,1}$ from the 11d Poincaré group $Iso(10,1)$ to the product group
(meaning that this subgroup is the stabilizer subgroup of the embedding).
Now consider instead super Minkowski spacetime $\mathbb{R}^{10,1\vert \mathbf{32}}$ (with $\mathbf{32}$ the irreducible Majorana spinor representation in 11), hence the local model superspace for super spacetimes in 11-dimensional supergravity. We are to ask what subspace of the spin representation $\mathbf{32}$ preserves the embedding in that the spinor bilinear pairing $\overline{\psi}_1 \Gamma \psi_2$ on that subspace lands in $\mathbb{R}^{2,1} \hookrightarrow Iso(2,1) \hookrightarrow Iso(10,1)$ (Sorokin 99, section 5.1). This is found to be the case for a half-dimensional subspace, and hence we may lift the above to a super-embedding of the form
(where now $\mathbf{2}$ is the irreducible Majorana spinor representation in 3d, and $8 \otimes \mathbf{2}$ denotes the direct sum of 8 copies of it) such that the induced stabilizer supergroup inside the super Poincaré group now is
It is in this sense that the membrane “breaks exactly half the supersymmetry”, namely from $\mathbf{32}$ to $8 \otimes \mathbf{2}$.
If one now thinks of this not as inclusions of global spacetimes, but of their super tangent spaces at the points where the membrane sits in spacetime, then this reflects the local structure of $\kappa$-symmetry: the $\kappa$-symmetries are locally generated by the 16 odd dimensions in $Iso(\mathbb{R}^{2,1\vert 8 \otimes \mathbf{2}} )$, being super-translations along the membrane worldvolume.
This explains why $\kappa$-symmetry in Green-Schwarz sigma models is taken to quotient out precisely half the spinor components, hence why, in the fully super-covariant formulation, one takes the worldvolume of a super $p$-brane in a superspacetime locally modeled on $\mathbb{R}^{d-1,1\vert \mathbf{N}}$ to be $\mathbb{R}^{p,1\vert \mathbf{N}/2}$. But notice that this is not a mathematical necessity. One may consider the worldvolume instead to have fewer odd directions. This then describes sigma models for “non-BPS super $p$-branes” (or rather “non-half-BPS” ).
Historically $\kappa$-symmetry was originally observed and considered for Green-Schwarz sigma models whose worldvolume is regarded as an ordinary (bosonic) smooth manifold. Then $\kappa$-symmetry is a “hidden” symmetry, with no evident geometric interpretation. As such it was first observed for the super-particle (Azcarraga-Lukierski 82, Siegel 83) and then for the super 1-brane in 3d (Siegel 84).
Based on this the Green-Schwarz action functional for the superstring in 10d (heterotic string, type II superstring) was found in (Green-Schwarz 84) by first observing that the plain Nambu-Goto action for a string on a supermanifold has twice as many fermionic on-shell degrees of freedom as the NSR-string and then by adding a term to the action (the WZW-term) to correct this defect, by ensuring that the sum of the NG-action with the WZW term enjoys $\kappa$-symmetry. By the same recipe later $\kappa$-symmetric Green-Schwarz-type sigma-model actions for all the other super p-branes were found, for instance for the M2-brane in (Bergshoeff-Sezgin-Townsend 87).
Originally, $\kappa$-symmetry was observed for the super-particle in
J. de Azcárraga, J. Lukierski, Supersymmetric particles, Phys. Lett. B113 (1982) 170; Phys. Rev. D28 (1983) 1337.
Warren Siegel, Hidden Local Supersymmetry In The Supersymmetric Particle Action Phys. Lett. B 128, 397 (1983)
and for the super 1-brane in 3d in
Then it was used to define/construct the manifestly spacetime supersymmetric Green-Schwarz action functional for the superstring in 10d in
and then for the M2-brane in 11d in
and so forth (see the references at super p-brane).
The super-geometric interpretation of $\kappa$-symmetry as the odd-graded part of the action of super-diffeomorphism on the super p-brane worldvolume, regarded itself as a supermanifold was first suggested in
Review of this perspective includes
Dmitri Sorokin, Superbranes and Superembeddings, Phys.Rept.329:1-101,2000 (arXiv:hep-th/9906142)
Paul Howe, Ergin Sezgin, section 4.3 of The supermembrane revisited, Class.Quant.Grav. 22 (2005) 2167-2200 (arXiv:hep-th/0412245)
Last revised on September 22, 2017 at 12:38:54. See the history of this page for a list of all contributions to it.