I am confused with morphisms of supermanifolds. Take a simple example $f:R^{0|1}\to R^{0|1}$. By (one of) definition, $f$ is a morphism of superalgebras of functions $C(R^{0|1})\to C(R^{0|1})$. Morphisms of superalgebras preserve the grading, I deduced that $f$ have the form $1\mapsto 1, \theta\mapsto x\theta$, i.e. $Hom(R^{0|1}\to R^{0|1})=R^1$ (as a set?). But I read from a paper that $Hom(R^{0|1},R^{0|1})=R^{1|1}$. What is going on? Thanks in advance!
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2 Answers
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You are right that the set of supermanifold morphisms $Hom(\mathbb R^{0|1},\mathbb R^{0|1})$ to itself is $\mathbb R^1$. However, one can define for supermanifolds $X,Y$ with $\dim X=0|d$ a supermanifold $map(X,Y)$ of morphisms from $X$ to $Y$, by $Hom(Z,map(X,Y))=Hom(Z\times X,Y)$ for all supermanifolds $Z$.
And $map(\mathbb R^{0|1},\mathbb R^{0|1})=\mathbb R^{1|1}$.
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$\begingroup$ Very interesting! I reread the paper, it says actrually inter-hom, maybe it your $map$. Should it be $X=R^{0|d}$? $\endgroup$– Ma MingCommented Feb 10, 2011 at 1:39
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1$\begingroup$ @Ma: Yes, the inner hom is the same thing as map. Which paper are you reading, by the way? $\endgroup$ Commented Feb 10, 2011 at 2:26
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1$\begingroup$ As @Ma Ming said, it suffices that $X$ be $0|\delta$-dimensional for $\operatorname{Maps}(X,Y)$ to be finite-dimensional. However, it's not good enough for $Y$ to be $0|\delta$-dimensional. For example, $\operatorname{Hom}(\mathbb R^{1|1},\mathbb R^{0|1}) = \operatorname{Hom}(\mathbb R[\epsilon], \mathcal C^\infty(\mathbb R)[\epsilon])$, where $\epsilon^2 = 0$ and $\epsilon$ is in odd degree; but such a map is any $\epsilon \mapsto f(x)\epsilon$, so this hom space is infinite-dimensional $\mathcal C^\infty(\mathbb R)$. $\endgroup$ Commented Feb 10, 2011 at 2:55
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1$\begingroup$ Thank you all! Could you give a clue how to calculate the $map$, for example, $map(R^{0|d},M)$ for any supermanifold $M$? $\endgroup$– Ma MingCommented Feb 10, 2011 at 11:49
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@Ma: As an answer to your following question:
Could you give a clue how to calculate the $map$, for example, $map(R^{0∣d},M)$ for any supermanifold $M$?
Take a look at arXiv:math/0307303, where this question is discussed.
For $d=1$, it is well-known (and due to Kontsevich, I think), that $map(R^{0|1},M)$ is the total space of the odd tangent bundle $\Pi TM$ of $M$.
If $U$ is a superdomain of dimension $p|q$, then $map(R^{0|d},U)=U\times R^{pr+qs|ps+qr}$ where $(r+1)|s=2^{d-1}|2^{d-1}$ is the graded dimension of $\bigwedge R^d$. This you can check using the definition of $map$ and the characterisation of morphisms of supermanifolds as given in Leites.
Another good source on this subject (for $d=1$), is the paper "Differential forms and 0-dimensional supersymmetric field theories" by Hohnhold, Kreck, Stolz and Teichner.
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$\begingroup$ Clickable arxiv:math/0307303: Kochan and Severa - Differential gorms, differential worms (not a typo, I think!). $\endgroup$– LSpiceCommented Dec 23, 2019 at 1:42