nLab
super Euclidean group

Idea

The ordinary Euclidean group? of ${ℝ}^n$ is the group generated from the rigid translation action of ${ℝ}^n$ on itself and rotations about the origin.

The super Euclidean group is analogously the supergroup of translations and rotations of the supermanifold ${ℝ}^{p\mid q}$.

Its super Lie algebra should be the super Poincare Lie algebra (up to the signature of the metric).

Details

incomplete for the moment, to be finished off tomorrow

The following description of the super Euclidean group (once it is finished, and polished) is due to Stephan Stolz and Peter Teichner.

The data needed to define the super Euclidean group is

1. $V$ a $d$-dimensional inner product space

2. a spinor representation ${\Delta }^{*}$ of $\mathrm{Spin}(V)$

3. a $\mathrm{Spin}(V)$-equivariant map

   $$\Gamma :{\Delta }^{*}{\otimes }_{ℂ}{\Delta }^{*}\to V{\otimes }_{ℝ}ℂ$$\Gamma : \Delta^* \otimes_{\mathbb{C}} \Delta^* \to V \otimes_{\mathbb{R}} \mathbb{C}

where $\mathrm{Spin}(V)$ is the Spin group (see Clifford algebra for the moment).

Here is the construction of $\mathrm{Eucl}({ℝ}^{d\mid \delta })$ for

• $d={\dim }_{ℝ}V$

remark $\delta$ is a multiple of $2^{[\frac{d-1}{2}]}$

set

$$X=\Pi \left(\begin{array}{c}V\times {\Delta }^{*}\\ \downarrow \\ V\end{array}\right)=V\times \Pi {\Delta }^{*}$$X = \Pi \left( \array{ V \times \Delta^* \\ \downarrow \\ V} \right) = V \times \Pi \Delta^*

is a complex supermanifold of dimension $(d\mid \delta )$

$$C^{\mathrm{\infty }}(V\times \Pi {\Delta }^{*})=C^{\mathrm{\infty }}(V)\otimes {\wedge }^{•}\Delta =C^{\mathrm{\infty }}(V,{\wedge }^{•}(\Delta ))$$C^\infty(V \times \Pi \Delta^*) = C^\infty(V) \otimes \wedge^\bullet \Delta = C^\infty(V, \wedge^\bullet(\Delta))

for $\delta =1$ this is

$$\cdots =C^{\mathrm{\infty }}(C,{\wedge }^{•}\Delta {)}^{\mathrm{ev}}\oplus C^{\mathrm{\infty }}(C,{\wedge }^{•}\Delta {)}^{\mathrm{odd}}$$\cdots = C^\infty(C, \wedge^\bullet \Delta)^{ev} \oplus C^\infty(C, \wedge^\bullet \Delta)^{odd}

$$\cdots \simeq C^{\mathrm{\infty }}(V)\oplus C^{\mathrm{\infty }}(V,{\wedge }^{•}\Delta )$$\cdots \simeq C^\infty(V) \oplus C^\infty(V, \wedge^\bullet \Delta)

where the last factor is $\simeq C^{\mathrm{\infty }}(V;\Delta )\simeq C^{\mathrm{\infty }}(V,S^{+})$ where $S^{+}$ is the spinor bundle

now define the multiplication

$$(V\times \Pi {\Delta }^{*})\times (V\times \Pi {\Delta }^{*})\stackrel{\mu }{\to }(V\times \Pi {\Delta }^{*})$$(V \times \Pi \Delta^*) \times (V \times \Pi \Delta^*) \stackrel{\mu}{\to} (V \times \Pi \Delta^*)

by sayin what it does on sets of probes by $S$

$$(V\times \Pi {\Delta }^{*})(S)\times (V\times \Pi {\Delta }^{*})(S)\stackrel{\mu (S)}{\to }(V\times \Pi {\Delta }^{*})(S)$$(V \times \Pi \Delta^*)(S) \times (V \times \Pi \Delta^*)(S) \stackrel{\mu(S)}{\to} (V \times \Pi \Delta^*)(S)

here on the left we have the sets of sections

$$C^{\mathrm{\infty }}(S{)}^{\mathrm{ev}}\otimes V\times C^{\mathrm{\infty }}(S{)}^{\mathrm{ev}}\otimes {\Delta }^{*}$$C^\infty(S)^{ev} \otimes V \times C^\infty(S)^{ev} \otimes \Delta^*

so we can map these as

$$((v_1,{\theta }_1),(v_2,{\theta }_2))\mapsto (v_1+v_2+\Gamma ({\theta }_1\otimes {\theta }_2),{\theta }_1+{\theta }_2)$$((v_1, \theta_1), (v_2, \theta_2) ) \mapsto (v_1 + v_2 + \Gamma(\theta_1 \otimes \theta_2), \theta_1 + \theta_2)

Remark

if the data $(V,{\Delta }^{*},\Gamma )$ and $(V\prime ,({\Delta }^{*})\prime ,\Gamma \prime )$ is isomorphic we get compatible notions of structures

But if $d=0,1,2$ and $\delta =1$ then there is a unique such triple with non-degenerate pairing $\Gamma$ up to isomorphism.

Definition

The structure of a Euclidean supermanifold on a $(d\mid \delta )$-dimensional supermanifold $Y$ is a $(V\times \Pi {\Delta }^{*},\mathrm{End}(V,{\Delta }^{*},\Gamma ))$-structure. See there for details.

Examples

recall the Clifford algebra table:

$$\begin{array}{ccc}d& \mathrm{Cl}({ℝ}^d{)}^{\mathrm{ev}}& \mathrm{Spin}({ℝ}^d)\\ \\ 1& ℝ& \{\pm 1\}\\ 2& (ℝ1\oplus ℝe_1{e}_2,e_i^2=1)\simeq ℂ& S^1\end{array}$$\array{ d & Cl(\mathbb{R}^d)^{ev} & Spin(\mathbb{R}^d) \\ \\ 1 & \mathbb{R} & \{\pm 1\} \\ 2 & (\mathbb{R} 1 \oplus \mathbb{R} e_1 e_2, e_i^2 = 1) \simeq \mathbb{C} & S^1 }

the group structure on $V\times \Pi {\Delta }^{*}$ is that of the “translations” and “rotations”

it will be defined on generalized elements with domain $S$ by maps of sets

$$\mu :(V\times \Pi {\Delta }^{*})(S)\times (V\times \Pi {\Delta }^{*})(S)\to (V\times \Pi {\Delta }^{*})(S)$$\mu: (V \times \Pi \Delta^*)(S) \times (V \times \Pi \Delta^*)(S) \to (V \times \Pi \Delta^*)(S)

$$(v_1,{\theta }_1),(v_2,{\theta }_2)\mapsto (v_1+v_2+\Gamma ({\theta }_1\otimes {\theta }_2),{\theta }_1+{\theta }_2)$$(v_1, \theta_1) , (v_2, \theta_2) \mapsto (v_1+ v_2 + \Gamma(\theta_1 \otimes \theta_2), \theta_1 + \theta_2)

$d=1$

${\Delta }^{*}=ℂ$

$$\Gamma :{\Delta }^{*}\otimes {\Delta }^{*}\to V{\otimes }_{ℝ}ℂ$$\Gamma : \Delta^* \otimes \Delta^* \to V \otimes_{\mathbb{R}} \mathbb{C}

$$ℂ\otimes ℂ\to ℝ{\otimes }_{ℝ}ℂ$$\mathbb{C} \otimes \mathbb{C} \to \mathbb{R} \otimes_{\mathbb{R}} \mathbb{C}

$$1\otimes 1\mapsto 1\otimes 1$$1 \otimes 1 \mapsto 1 \otimes 1

so here this is the super translation group.

$d=2$

${\Delta }^{*}=ℂ$

$$u\in S^1\simeq U(1)\simeq \mathrm{Spin}({ℝ}^2)$$u \in S^1 \simeq U(1) \simeq Spin(\mathbb{R}^2)

$${\Delta }^{*}{\otimes }_{ℂ}{\Delta }^{*}\stackrel{\Gamma }{\to }{ℝ}^2\otimes ℝ\simeq ℂ\oplus ℂ$$\Delta^* \otimes_{\mathbb{C}} \Delta^* \stackrel{\Gamma}{\to} \mathbb{R}^2 \otimes \mathbb{R} \simeq \mathbb{C} \oplus \mathbb{C}

the first map is multiplication by $u^{-1}$ and then the isomorphism on the right sends

$$(x,y)\otimes 1\mapsto (z,\overline{z})$$(x,y)\otimes 1 \mapsto (z, \bar z)

where $z=x+iy$

translation group $V\times \Pi {\Delta }^{*}\simeq {ℝ}^{2\mid 1}$

multiplication on $S$-elements

$${ℝ}^{2\mid 1}(S)\times {ℝ}^{2\mid 1}(S)\to {ℝ}^{2\mid 1}(S)$$\mathbb{R}^{2|1}(S) \times \mathbb{R}^{2|1}(S) \to \mathbb{R}^{2|1}(S)

given by

$$(z_1,{\overline{z}}_1,{\theta }_1),(z_2,{\overline{z}}_2,{\theta }_2)\mapsto (z_1+z_2,{\overline{z}}_1+{\overline{z}}_2+{\theta }_1{\theta }_2,{\theta }_1+{\theta }_2)$$(z_1,\bar z_1, \theta_1), (z_2,\bar z_2, \theta_2) \mapsto (z_1 + z_2, \bar z_1 + \bar z_2 + \theta_1 \theta_2, \theta_1 + \theta_2)