nLab
Euclidean supermanifold

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

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

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

• a maximal atlas consisting of charts

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

  (where on the left $Y_{red}{\supset }_{\mathrm{open}}(U_i{)}_{\mathrm{red}}$) with $O_Y{\mid }_{(U_i{)}_{\mathrm{red}}=O_{U_i}}$

• such that the transition function

  $$X\supset {\varphi }_i(U_i\cap U_j)\stackrel{{\varphi }_j\circ {\varphi }_i^{-1}}{\to }{\varphi }_j(U_i\cap U_j)\subset X$$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

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

family version

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

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

together with a maximal atlas of charts

$$\begin{array}{ccccc}Y{\supset }_{\mathrm{open}}{U}_i& & \stackrel{{\varphi }_i}{{\to }_{\simeq }}& & V_i{\subset }_{\mathrm{open}}S\times X\\ & \searrow & & \swarrow \\ & & S\end{array}$$\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

$$\begin{array}{ccccc}S\times X\supset {\varphi }_i(U_i\times U_j)& & \to & & {\varphi }_j(U_i\cap U_j)\subset S\times X\\ & \searrow & & \swarrow \\ & & S\end{array}$$\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

$$S\times X\stackrel{\mathrm{Id}\times g\times \mathrm{id}}{\to }S\times G\times X\stackrel{\mathrm{Id}\times \mathrm{action}}{\to }$$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 $({ℝ}^d,\mathrm{Eucl}({ℝ}^d))$-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 $({ℝ}^2,{ℝ}^2⋊\mathrm{Spin}(2))$-structure.

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

In general for two structures $(X,G)$ and $(X\prime ,G\prime )$ 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 $(X,G)$-chart to the corresponding $(X\prime ,G\prime )$-chart which as a subset of $X\prime$ is the inverse image of $X\prime \to X$.

This yields a functor

$$(X,G)-\mathrm{manifolds}\to (X\prime ,G\prime )-\mathrm{manifolds}.$$(X,G)-manifolds \to (X',G')-manifolds \,.