nLab super-translation group

Idea

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

Definition

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

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

See at super Minkowski spacetime.

The underlying super vector space of this is

$$\mathbb{R}^d \oplus \Pi S \,,$$

where $S$ 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:

$$[\psi, \phi] = \langle \psi, \Gamma^a \phi \rangle t_a \,,$$

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

Properties

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 $\mathbb{R}^{0;N}$ by the $d$ super Lie algebra cocycles

$$(\phi, \psi) \mapsto \langle \phi, \Gamma^a \psi\rangle t_a \,,$$

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

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 $\mathbb{R}^{11;N=1}$ as a central extension of $\mathbb{R}^{10;N=(1,1)}$, 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).

Examples

In dimension 1

The additive group structure on $\mathbb{R}^{1|1}$ 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

$$\mathbb{R}^{1|1} \times \mathbb{R}^{1|1} \to \mathbb{R}^{1|1}$$

$$(t_1, \theta_1), (t_2, \theta_2) \mapsto (t_1 + t_2 + \theta_1 \theta_2, \theta_1 + \theta_2) \,.$$

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

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

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

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

$$\mathbb{R}^{1|1}(S) \times \mathbb{R}^{1|1}(S) \to \mathbb{R}^{1|1}(S)$$

$$(t_1, \theta_1), (t_2, \theta_2) \mapsto (t_1 + t_2 + \theta_1 \theta_2, \theta_1 + \theta_2) \,.$$

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

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

Related concepts

• translation Lie algebra

• super Minkowski spacetime

• super Poincaré group

• super Poincaré Lie algebra

References

• Veeravalli Varadarajan, section 7 of Supersymmetry for mathematicians: An introduction

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

• C. Chryssomalakos, José de Azcárraga, J.M. Izquierdo, J.C. Pérez Bueno, The geometry of branes and extended superspaces, Nucl.Phys.B567:293-330, 2000 (arXiv:hep-th/9904137)

and more generally in the context of The brane bouquet in

• Domenico Fiorenza, Hisham Sati, Urs Schreiber, Super Lie n-algebra extensions, higher WZW models and super p-branes with tensor multiplet fields