superalgebra

and

supergeometry

# Idea

A Euclidean supermanifold is a supermanifold that can be thought of as being equipped with a flat Riemannian metric.

Alternatively, it is a supermanifold for which the transition functions of an atlas are restricted to be elements of the super Euclidean group.

# Definition

A Euclidean supermanifold of dimension $\left(p\mid q\right)$is a supermanifold that is quipped with an $\left(X,G\right)$-structure , where $X={ℝ}^{p\mid q}$ and where $G$ is the super Euclidean group on ${ℝ}^{p\mid q}$.

Here an $\left(X,G\right)$-structure is defined as follows, essentially being a version of the discussion of pseudogroups at manifold.

Definition (Stolz, Teichner) A $\left(X,G\right)$-structure on a $\left(d\mid \delta \right)$-dimensional supermanifold $Y$ consists of

• a maximal atlas consisting of charts

$Y\underset{\mathrm{opn}}{\mathrm{sup}}{U}_{i}\stackrel{{\varphi }_{i}}{{\to }_{\simeq }}{V}_{i}{\subset }_{\mathrm{open}}X$Y \sup_{opn} U_i \stackrel{\phi_i}{\to_\simeq} V_i \subset_{open} X

(where on the left ${Y}_{red}{\supset }_{\mathrm{open}}\left({U}_{i}{\right)}_{\mathrm{red}}$) with ${O}_{Y}{\mid }_{\left({U}_{i}{\right)}_{\mathrm{red}}={O}_{{U}_{i}}}$

• such that the transition function

$X\supset {\varphi }_{i}\left({U}_{i}\cap {U}_{j}\right)\stackrel{{\varphi }_{j}\circ {\varphi }_{i}^{-1}}{\to }{\varphi }_{j}\left({U}_{i}\cap {U}_{j}\right)\subset X$X \supset \phi_i(U_i \cap U_j) \stackrel{\phi_j \circ \phi_i^{-1}}{\to} \phi_j(U_i \cap U_j) \subset X

is the restriction of a map

$X\simeq X×\mathrm{pt}\stackrel{\mathrm{id}×g}{\to }X×G\stackrel{\mathrm{action}}{\to }X$X \simeq X \times pt \stackrel{id \times g}{\to} X \times G \stackrel{action}{\to} X

## family version

definition A family of $\left(X,G\right)$-(complex-, super-)manifolds is a map

$\begin{array}{c}Y\\ {↓}^{p}\\ S\end{array}$\array{ Y \\ \downarrow^p \\ S }

together with a maximal atlas of charts

$\begin{array}{ccccc}Y{\supset }_{\mathrm{open}}{U}_{i}& & \stackrel{{\varphi }_{i}}{{\to }_{\simeq }}& & {V}_{i}{\subset }_{\mathrm{open}}S×X\\ & ↘& & ↙\\ & & S\end{array}$\array{ Y \supset_{open} U_i &&\stackrel{\phi_i}{\to_\simeq}&& V_i \subset_{open} S \times X \\ & \searrow && \swarrow \\ && S }

such that the transition maps

$\begin{array}{ccccc}S×X\supset {\varphi }_{i}\left({U}_{i}×{U}_{j}\right)& & \to & & {\varphi }_{j}\left({U}_{i}\cap {U}_{j}\right)\subset S×X\\ & ↘& & ↙\\ & & S\end{array}$\array{ S \times X \supset \phi_i(U_i \times U_j) &&\to&& \phi_j(U_i \cap U_j) \subset S \times X \\ & \searrow && \swarrow \\ && S }

are the restriction of a map of the following form

$S×X\stackrel{\mathrm{Id}×g×\mathrm{id}}{\to }S×G×X\stackrel{\mathrm{Id}×\mathrm{action}}{\to }$S\times X \stackrel{Id \times g \times id}{\to} S \times G \times X \stackrel{Id \times action}{\to}

for some

$S\stackrel{g}{\to }G$S \stackrel{g}{\to} G

example a family $Y\to S$ of $\left({ℝ}^{d},\mathrm{Eucl}\left({ℝ}^{d}\right)\right)$-manifolds, for $S$ an ordinary manifold is a submersion with flat Riemannian metric on the fibers.

# ordinary Euclidean manifolds as Euclidean supermanifolds

Specifically in 2-dimensions, an ordinary Spin-Eulidean manifold is one with $\left({ℝ}^{2},{ℝ}^{2}⋊\mathrm{Spin}\left(2\right)\right)$-structure.

We want to regard this as a Euclidean supermanifold with $\left({ℝ}_{\mathrm{cs}}^{2\mid 1},{ℝ}_{\mathrm{cs}}^{2\mid 1}⋊\mathrm{Spin}\left(2\right)\right)$-structure.

In general for two structures $\left(X,G\right)$ and $\left(X\prime ,G\prime \right)$ we can transfer structures when we have a group homomorphisms

$G\to G\prime$ and with respect to that a

$G$-equivariant map $X\prime \to X$.

Then send every $\left(X,G\right)$-chart to the corresponding $\left(X\prime ,G\prime \right)$-chart which as a subset of $X\prime$ is the inverse image of $X\prime \to X$.

This yields a functor

$\left(X,G\right)-\mathrm{manifolds}\to \left(X\prime ,G\prime \right)-\mathrm{manifolds}\phantom{\rule{thinmathspace}{0ex}}.$(X,G)-manifolds \to (X',G')-manifolds \,.
Revised on September 24, 2009 09:42:29 by Urs Schreiber (195.37.209.182)