nLab super-translation group




Group Theory



A super-translation group is a supergroup generalization of the translation group, hence of the additive Lie group n\mathbb{R}^n. Its underlying supermanifold is a super-Euclidean space or super-Minkowski spacetime.


Given a super Poincaré Lie algebra extension 𝔰𝔦𝔰𝔬 N(d1,1)\mathfrak{siso}_N(d-1,1) of a orthogonal Lie algebra 𝔰𝔬(d1,1)\mathfrak{so}(d-1,1) for some spin representation NN, then the corresponding super-translation Lie algebra is the quotient

d1,1|N𝔰𝔦𝔰𝔬 N(d1,1)/𝔬(d1,1) \mathbb{R}^{d-1,1\vert N} \coloneqq \mathfrak{siso}_N(d-1,1)/\mathfrak{o}(d-1,1)

See at super Minkowski spacetime.

The underlying super vector space of this is

dΠS, \mathbb{R}^d \oplus \Pi S \,,

where SS is the vector space underlying the given spin representation.

The super Lie algebra structure is mildly non-abelian with the only non-trivial bracket being that between two spinors and given by the bilinear pairing (the charge conjugation matrix) between two spinors:

[ψ,ϕ]=ψ,Γ aϕt a, [\psi, \phi] = \langle \psi, \Gamma^a \phi \rangle t_a \,,

where {t a}\{t_a\} is a basis for the translation generators in d\mathbb{R}^d.


As a central extension of the superpoint

The super-translation Lie algebra is a super-Lie algebra extension of the abelian super Lie algebra which is just the superpoint 0;N\mathbb{R}^{0;N} by the dd super Lie algebra cocycles

(ϕ,ψ)ϕ,Γ aψt a, (\phi, \psi) \mapsto \langle \phi, \Gamma^a \psi\rangle t_a \,,

for a{1,2,,d}a \in \{1, 2, \cdots, d\}, where {t a}\{t_a\} are the basis elements of d\mathbb{R}^d.

This simple but maybe noteworthy fact has been highlighted in the context of the brane scan in (CAIP 99, section 2.1).

This mechanism plays a role in string theory when realizing 11;N=1\mathbb{R}^{11;N=1} as a central extension of 10;N=(1,1)\mathbb{R}^{10;N=(1,1)}, for this formalizes aspects of the idea that type IIA string theory with a D0-brane condensate is 11-dimensional supergravity/M-theory (FSS 13).


In dimension 1

The additive group structure on 1|1\mathbb{R}^{1|1} is given on generalized elements in (i.e. in the logic internal to) the topos of sheaves on the category SCartSp of cartesian superspaces by

1|1× 1|1 1|1 \mathbb{R}^{1|1} \times \mathbb{R}^{1|1} \to \mathbb{R}^{1|1}
(t 1,θ 1),(t 2,θ 2)(t 1+t 2+θ 1θ 2,θ 1+θ 2). (t_1, \theta_1), (t_2, \theta_2) \mapsto (t_1 + t_2 + \theta_1 \theta_2, \theta_1 + \theta_2) \,.

Recall how the notation works here: by the Yoneda embedding we have a full and faithful functor

SDiff\hookrightarrow Fun(SDiff op,Set)Fun(SDiff^{op}, Set)

and we also have the theorem, discussed at supermanifolds, that maps from some SSDiffS \in SDiff into p|q\mathbb{R}^{p|q} is given by a tuple of pp even section t it_i and qq odd sections θ j\theta_j. The above notation specifies the map of supermanifolds by displaying what map of sets of maps from some test object SS it corresponds to under the Yoneda embedding.

Now, or each SS \in SDiff there is a group structure on the hom-set SDiff(S, 1|1)C (S) ev×C (X) oddSDiff(S, \mathbb{R}^{1|1}) \simeq C^\infty(S)^{ev} \times C^\infty(X)^{odd} given by precisely the above formula for this given SS

1|1(S)× 1|1(S) 1|1(S) \mathbb{R}^{1|1}(S) \times \mathbb{R}^{1|1}(S) \to \mathbb{R}^{1|1}(S)
(t 1,θ 1),(t 2,θ 2)(t 1+t 2+θ 1θ 2,θ 1+θ 2). (t_1, \theta_1), (t_2, \theta_2) \mapsto (t_1 + t_2 + \theta_1 \theta_2, \theta_1 + \theta_2) \,.

where (t i,θ i)C (S) ev×C (S) odd(t_i, \theta_i) \in C^\infty(S)^{ev} \times C^\infty(S)^{odd} etc and where the addition and product on the right takes place in the function super algebra C (S)C^\infty(S).

Since the formula looks the same for all SS, one often just writes it without mentioning SS as above.


Discussion in the context of the brane scan is in section 2.1 of

and more generally in the context of The brane bouquet in

Last revised on September 8, 2023 at 07:52:43. See the history of this page for a list of all contributions to it.