nLab
Euclidean supermanifold

Contents

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

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

Definition (Stolz, Teichner) A (X,G)(X,G)-structure on a (d|δ)(d|\delta)-dimensional supermanifold YY consists of

  • a maximal atlas consisting of charts

    Ysup opnU i ϕ iV i openX Y \sup_{opn} U_i \stackrel{\phi_i}{\to_\simeq} V_i \subset_{open} X

    (where on the left Y red open(U i) redY_\red\supset_{open} (U_i)_{red}) with O Y| (U i) red=O U iO_Y|_{(U_i)_{red} = O_{U_i}}

  • such that the transition function

    Xϕ i(U iU j)ϕ jϕ i 1ϕ j(U iU j)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

    XX×ptid×gX×GactionX 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)(X,G)-(complex-, super-)manifolds is a map

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

together with a maximal atlas of charts

Y openU i ϕ i V i openS×X S \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

S×Xϕ i(U i×U j) ϕ j(U iU j)S×X S \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×XId×g×idS×G×XId×action S\times X \stackrel{Id \times g \times id}{\to} S \times G \times X \stackrel{Id \times action}{\to}

for some

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

example a family YSY \to S of ( d,Eucl( d))(\mathbb{R}^d, Eucl(\mathbb{R}^d))-manifolds, for SS 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 ( 2, 2Spin(2))(\mathbb{R}^2, \mathbb{R}^2 \rtimes Spin(2))-structure.

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

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

GGG \to G' and with respect to that a

GG-equivariant map XXX' \to X.

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

This yields a functor

(X,G)manifolds(X,G)manifolds. (X,G)-manifolds \to (X',G')-manifolds \,.
Revised on September 24, 2009 09:42:29 by Urs Schreiber (195.37.209.182)