Within the algebrogeometric approach to supergeometry, a supermanifold of dimension $m|n$ is an ordinary $m$ dimensional smooth manifold $M$ and a sheaf of supercommutative super algebras $\mathbf{C}^\infty$ on $M$ (thought of as the sheaf of functions on the supermanifold) which is locally isomorphic to $C^\infty_M\otimes \bigwedge^* \mathbb{R}^n$.

The space underlying $\mathbb{R}^{m|n}$ is $\mathbb{R}^m$ and the sheaf of functions is $U\mapsto C^\infty(U)\otimes \bigwedge^* \mathbb{R}^n$. This means that $\mathbb{R}^{0|2}=(\mathbb{R}^0, C^\infty_{\mathbb{R}^0}\otimes \bigwedge^{*}\mathbb{R}^2)\cong (\ast, \bigwedge^*\mathbb{R}^2)$.

We can consider a trivial supermanifold structure on any smooth manifold by taking $\mathbf{C}_M^\infty=C_M^\infty\oplus 0$.

We can define a map between supermanifolds $(M, \mathbf{C}^\infty_M)\to (N, \mathbf{C}^\infty_N)$ as a smooth map $f:M\to N$ and a morphism $f^{*}: \mathbf{C}^\infty_N\to f_{*}\mathbf{C}^\infty_M$ where $f_{*}\mathbf{C}^\infty_M$ denotes the pushforward sheaf of $\mathbf{C}^\infty_M$ with respect to $f$, $(f_*\mathbf{C}^\infty_M)(U):=\mathbf{C}^\infty_M(f^{-1}(U))$.

As defined, a map $f:\mathbb{R}^{0|2}\to M$ between the superplane $\mathbb{R}^{0|2}$ and a smooth manifold $M$ consists of a map $\ast\to M$ and a morphism of sheaves of superalgebras $C^\infty_M\to f_*\bigwedge^{*}\mathbb{R}^2$ where $f_{*}\bigwedge^{*}\mathbb{R}^2(U)=0$ if $f(*)\notin U$ and $f_{*}\bigwedge^{*}\mathbb{R}^2(U)=\bigwedge^{*}\mathbb{R}^2$.

Since the sheaf $f_{*}\bigwedge^{*}\mathbb{R}^2$ is concentrated at $f(\ast)$ this map is uniquely determined by the map between stalks $C^\infty_{f(p)}\to \bigwedge^{*}\mathbb{R}^2$. Is there a nice way of understanding this space of maps?

(Migrated from Math Stack Exchange)

  • $\begingroup$ It seems that in your definition of $f^*$, the roles of $M$ and $N$ are swapped. $\endgroup$ Nov 4, 2022 at 12:34

1 Answer 1


The manifold $\def\Hom{\mathop{\rm Hom}}\def\R{{\bf R}}\Hom(\R^{0|2},M)$ is isomorphic to the pullback of the parity-reversed bundle vector bundle $TM⊕TM$ along the projection map $TM→M$.

This is Lemma 1.6 in Berwick-Evans's The Chern-Gauss-Bonnet theorem via supersymmetric Euclidean field theories.


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