superalgebra

and

supergeometry

Contents

Idea

A super-translation group is a supergroup generalization of the translation group, hence of the additive Lie group ${ℝ}^{n}$. Its underlying supermanifold is super-Minkowki spacetime?.

Definition

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

${ℝ}^{d;N}≔{\mathrm{𝔰𝔦𝔰𝔬}}_{N}\left(d-1,1\right)/𝔬\left(d-1,1\right)\phantom{\rule{thinmathspace}{0ex}}.$\mathbb{R}^{d;N} \coloneqq \mathfrak{siso}_N(d-1,1)/\mathfrak{o}(d-1,1) \,.

The underlying super vector space of this is

${ℝ}^{d}\oplus \Pi S\phantom{\rule{thinmathspace}{0ex}},$\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:

$\left[\psi ,\varphi \right]=⟨\psi ,{\Gamma }^{a}\varphi ⟩{t}_{a}\phantom{\rule{thinmathspace}{0ex}},$[\psi, \phi] = \langle \psi, \Gamma^a \phi \rangle t_a \,,

where $\left\{{t}_{a}\right\}$ is a basis for the translation generators in ${ℝ}^{d}$.

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

$\left(\varphi ,\psi \right)↦⟨\varphi ,{\Gamma }^{a}\psi ⟩{t}_{a}\phantom{\rule{thinmathspace}{0ex}},$(\phi, \psi) \mapsto \langle \phi, \Gamma^a \psi\rangle t_a \,,

for $a\in \left\{1,2,\cdots ,d\right\}$, where $\left\{{t}_{a}\right\}$ are the basis elements of ${ℝ}^{d}$.

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

${ℝ}^{1\mid 1}×{ℝ}^{1\mid 1}\to {ℝ}^{1\mid 1}$\mathbb{R}^{1|1} \times \mathbb{R}^{1|1} \to \mathbb{R}^{1|1}
$\left({t}_{1},{\theta }_{1}\right),\left({t}_{2},{\theta }_{2}\right)↦\left({t}_{1}+{t}_{2}+{\theta }_{1}{\theta }_{2},{\theta }_{1}+{\theta }_{2}\right)\phantom{\rule{thinmathspace}{0ex}}.$(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 $↪$ $\mathrm{Fun}\left({\mathrm{SDiff}}^{\mathrm{op}},\mathrm{Set}\right)$

and we also have the theorem, discussed at supermanifolds, that maps from some $S\in \mathrm{SDiff}$ into ${ℝ}^{p\mid q}$ 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 $\mathrm{SDiff}\left(S,{ℝ}^{1\mid 1}\right)\simeq {C}^{\infty }\left(S{\right)}^{\mathrm{ev}}×{C}^{\infty }\left(X{\right)}^{\mathrm{odd}}$ given by precisely the above formula for this given $S$

${ℝ}^{1\mid 1}\left(S\right)×{ℝ}^{1\mid 1}\left(S\right)\to {ℝ}^{1\mid 1}\left(S\right)$\mathbb{R}^{1|1}(S) \times \mathbb{R}^{1|1}(S) \to \mathbb{R}^{1|1}(S)
$\left({t}_{1},{\theta }_{1}\right),\left({t}_{2},{\theta }_{2}\right)↦\left({t}_{1}+{t}_{2}+{\theta }_{1}{\theta }_{2},{\theta }_{1}+{\theta }_{2}\right)\phantom{\rule{thinmathspace}{0ex}}.$(t_1, \theta_1), (t_2, \theta_2) \mapsto (t_1 + t_2 + \theta_1 \theta_2, \theta_1 + \theta_2) \,.

where $\left({t}_{i},{\theta }_{i}\right)\in {C}^{\infty }\left(S{\right)}^{\mathrm{ev}}×{C}^{\infty }\left(S{\right)}^{\mathrm{odd}}$ etc and where the addition and product on the right takes place in the function super algebra ${C}^{\infty }\left(S\right)$.

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

References

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

Revised on August 30, 2013 17:40:09 by Urs Schreiber (89.204.154.236)