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

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 $(d|\delta)$-dimensional supermanifold $Y$ consists of

• a maximal atlas consisting of charts

  $$Y \sup_{opn} U_i \stackrel{\phi_i}{\to_\simeq} V_i \subset_{open} X$$

  (where on the left $Y_\red\supset_{open} (U_i)_{red}$) with $O_Y|_{(U_i)_{red} = O_{U_i}}$

• such that the transition function

  $$X \supset \phi_i(U_i \cap U_j) \stackrel{\phi_j \circ \phi_i^{-1}}{\to} \phi_j(U_i \cap U_j) \subset X$$

  is the restriction of a map

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

family version

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

$$\array{ Y \\ \downarrow^p \\ S }$$

together with a maximal atlas of charts

$$\array{ Y \supset_{open} U_i &&\stackrel{\phi_i}{\to_\simeq}&& V_i \subset_{open} S \times X \\ & \searrow && \swarrow \\ && S }$$

such that the transition maps

$$\array{ S \times X \supset \phi_i(U_i \times U_j) &&\to&& \phi_j(U_i \cap U_j) \subset S \times X \\ & \searrow && \swarrow \\ && S }$$

are the restriction of a map of the following form

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

for some

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

example a family $Y \to S$ of $(\mathbb{R}^d, Eucl(\mathbb{R}^d))$-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 $(\mathbb{R}^2, \mathbb{R}^2 \rtimes Spin(2))$-structure.

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

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

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

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

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

This yields a functor

$$(X,G)-manifolds \to (X',G')-manifolds \,.$$