# On $(k\oplus l|q)$-dimensional supermanifolds.

Anatoly Konechny and Albert Schwarz
Department of Mathematics, University of California,
Davis, CA 95616
<EMAIL_ADDRESS>
<EMAIL_ADDRESS>
(April 22, 1997)

###### Abstract

We define a $(k\oplus l|q)$-dimensional supermanifold as a manifold having $q$
odd coordinates and $k+l$ even coordinates with $l$ of them taking only
nilpotent values. We show that this notion can be used to formulate
superconformal field theories with different number of supersymmetries in
holomorphic and antiholomorphic sectors.

The present preprint is an extended version of paper [1]. It stems from an
attempt to understand the precise mathematical meaning of some constructions
used by physicists. We have in mind in particular the consideration of
independent spin structures in holomorphic and antiholomorphic sectors of N=1
superconformal field theory and the notion of chiral (heterotic) supermanifold
(for example see [2]). In this paper we introduce a notion of $(k\oplus
l|q)$-dimensional supermanifold and show that in many cases rigorous
definitions can be based on this notion. Mathematical details are relegated to
the appendix that will be published separately. The appendix also contains the
details of formulation of supergeometry in terms of functors. This language is
most suitable for our purposes and is convenient in many other supergeometry
questions as well.

Let $\Lambda=\Lambda_{0}+\Lambda_{1}$ be a Grassmann algebra with an even
subspace $\Lambda_{0}$ and an odd subspace $\Lambda_{1}$. The space ${\bf
R}_{\Lambda}^{p|q}$ can be defined as a space consisting of rows
$(x^{1},\dots,x^{p},\xi^{1},\dots,\xi^{q})$ where
$x^{1},\dots,x^{p}\in\Lambda_{0}$ are even elements of Grassmann algebra
$\Lambda$ and $\xi^{1},\dots,\xi^{q}\in\Lambda_{1}$ are odd elements from
$\Lambda$. Physicists usually say that one can take as $\Lambda$ any Grassmann
algebra provided it is large enough. From the viewpoint of a mathematician it
is better to consider a family of sets ${\bf R}_{\Lambda}^{p|q}$ corresponding
to all Grassmann algebras $\Lambda$. It is easy to see that a parity
preserving homomorphism $\alpha:\Lambda\rightarrow\Lambda^{\prime}$ generates
naturally a map $\tilde{\alpha}:{\bf R}_{\Lambda}^{p|q}\rightarrow{\bf
R}_{\Lambda^{\prime}}^{p|q}$ and that
$\widetilde{\beta\alpha}=\tilde{\beta}\tilde{\alpha}$ for any parity
preserving homomorphisms
$\alpha:\Lambda\rightarrow\Lambda^{\prime},\enspace\beta:\Lambda^{\prime}\rightarrow\Lambda^{\prime\prime}$.
In the language of mathematics this means that the correspondence
$\Lambda\mapsto{\bf R}_{\Lambda}^{p|q}$ determines a functor acting from the
category of Grassmann algebras into the category of sets (or of vector
spaces).

An (even) superfield on ${\bf R}^{p|q}$ can be defined as an expression of the
form

$\sum_{k=2l}\sum_{1\leq
i_{1}<\dots<i_{k}}f_{i_{1},\dots,i_{k}}(x^{1},\dots,x^{p})\xi^{i_{1}}\dots\xi^{i_{k}}$
(1)

Here $f_{i_{1},\dots,i_{k}}$ are smooth functions on ${\bf R}^{p}$. It is
important to notice that such an expression determines a map $F_{\Lambda}:{\bf
R}_{\Lambda}^{p|q}\rightarrow\Lambda_{0}={\bf R}_{\Lambda}^{1|0}$. This
follows from the fact that we can substitute an even element of a Grassmann
algebra into any smooth function of real variable. To verify this statement we
notice that every even element $x$ of Grassmann algebra can be represented in
the form $x=m+n$ where $m\in{{\bf R}}$ and $n$ is nilpotent. We define $f(x)$
using the Taylor expansion
$f(x)=\sum_{k=0}^{\infty}\frac{f^{(k)}(m)}{k!}n^{k}$ (the series terminates
because $n$ is nilpotent). If $\alpha:\Lambda\rightarrow\Lambda^{\prime}$ is a
parity preserving homomorphism then $\tilde{\alpha}\circ
F_{\Lambda}=F_{\Lambda}\circ\tilde{\alpha}$. This means in mathematical
terminology that a superfield (1) specifies a natural map of the functor ${\bf
R}^{p|q}$ into the functor ${\bf R}^{1|0}$. Analogously, an odd superfield can
be considered as a natural transformation of functors ${\bf
R}^{p|q}\rightarrow{\bf R}^{0|1}$. An arbitrary superfield can be viewed as a
natural transformation of functors ${\bf R}^{p|q}\rightarrow{\bf R}^{1|1}$
(recall that ${\bf R}^{1|1}_{\Lambda}=\Lambda_{0}\oplus\Lambda_{1}$ ). Note
that in the above considerations the Grassmann algebra can be replaced by any
algebra $\Lambda$ every element of which can be represented as a sum of a real
number and nilpotent element ( assuming that $\Lambda$ is associative,
$Z_{2}$-graded , supercommutative algebra having unit element). Algebras of
this kind will be called almost nilpotent algebras or AN algebras. In other
words we can say that ${\bf R}^{p|q}$ can be considered as a functor on the
category of AN algebras and a superfield of definite parity determines a
natural transformation of the functor ${\bf R}^{p|q}$ into the functor ${\bf
R}^{1|0}$ or ${\bf R}^{0|1}$.

The main notions of superalgebra and of supergeometry can be formulated very
easily in the language of functors. A superspace can be defined as an
arbitrary functor on the category $\bf AN$ of AN algebras taking values in the
category of sets. The body of a superspace can be defined as the set
corresponding to the AN algebra $\Lambda={\bf R}$. We introduce a
$(p|q)$-dimensional supermanifold as a superspace that is locally equivalent
to ${\bf R}^{p|q}$. In other words $(p|q)$-dimensional supermanifold can be
pasted together from domains in ${\bf R}^{p|q}$ by means of smooth
transformations. A body of a $(p|q)$-dimensional supermanifold is a
$p$-dimensional smooth manifold. Replacing in the definition of a superspace
the category of sets by the category of groups or by the category of Lie
algebras we obtain the definitions of supergroup and super Lie algebra
respectively.

It is convenient to generalize the notion of the superspace ${\bf R}^{p|q}$ as
follows. Denote by ${\bf R}^{k\oplus l|q}_{\Lambda}$ a set of rows
$(x^{1},\dots,x^{k},y^{1},\dots,y^{l},\xi^{1},\dots,\xi^{q})$ where
$x^{1},\dots,x^{k}$ are arbitrary even elements of AN algebra $\Lambda$ ,
$y^{1},\dots,y^{l}$ are nilpotent even elements of $\Lambda$ and
$\xi^{1},\dots,\xi^{q}$ are odd elements of $\Lambda$. A superspace ${\bf
R}^{k\oplus l|q}$ can be defined as a functor $\Lambda\rightarrow{\bf
R}^{k\oplus l|q}_{\Lambda}$. Strictly speaking to define a functor we should
also construct a homomorphism $\tilde{\alpha}:{\bf R}^{k\oplus
l|q}_{\Lambda}\rightarrow{\bf R}^{k\oplus l|q}_{\Lambda^{\prime}}$ for every
parity preserving homomorphism $\alpha:\Lambda\rightarrow\Lambda^{\prime}$ of
AN algebras. We omit this obvious construction.

Let us define a superfield on ${\bf R}^{k+l|q}$ as an expression of the form

$\sum_{s}\sum_{1\leq
i_{1}<\dots<i_{s}}f_{i_{1},\dots,i_{s}}(x^{1},\dots,x^{k},y^{1},\dots,y^{l})\xi^{i_{1}}\dots\xi^{i_{s}}$
(2)

where $f_{i_{1},\dots,i_{s}}$ are smooth functions of variables
$(x^{1},\dots,x^{k})\in{\bf R}^{k}$ and formal power series with respect to
$y^{1},\dots,y^{l}$. Such an expression determines a map of ${\bf R}^{k\oplus
l|q}$ into ${\bf R}^{1|0}$ if expression (2) is even and into ${\bf R}^{0|1}$
if it is odd.

We define a $(k\oplus l|q)$-dimensional supermanifold as a superspace that is
locally equivalent to ${\bf R}^{k\oplus l|q}$. Almost all notions of
(super)geometry can be generalized to the case of $(k\oplus l|q)$-dimensional
supermanifolds. In particular one can define the supergroup of transformations
of such a manifold and corresponding Lie (super) algebra of vector fields. As
usual, a vector field on a $(k\oplus l|q)$-dimensional superdomain with
coordinates $(x^{1},\dots,x^{k},y^{1},\dots,y^{l},\xi^{1},\dots,\xi^{q})$ can
be identified with a first order differential operator

$A^{i}\frac{\partial}{\partial x^{i}}+B^{j}\frac{\partial}{\partial
y^{j}}+C^{s}\frac{\partial}{\partial\xi^{s}}$ (3)

where $A^{i},B^{j},C^{s}$ are smooth functions of variables
$x^{1},\dots,x^{k}$ , formal power series with respect to $y^{1},\dots,y^{l}$
and polynomials in $\xi^{1},\dots,\xi^{q}$. If $A^{i},B^{j}$ are odd and
$C^{s}$ are even then the operator (3) is parity reversing ; we say that the
corresponding vector field is odd. The definition of an even vector field is
similar. A $(k\oplus l|q)$-dimensional supermanifold is pasted together from
$(k\oplus l|q)$-dimensional superdomains by means of smooth transformations.
If we have a $(p|q)$-dimensional supermanifold $\cal A$ and a $k$-dimensional
submanifold ${\cal B}^{(0)}$ of its body ${\cal A}^{(0)}$ we can construct
easily a $(k\oplus(p-k)|q)$-dimensional supermanifold in the following way.
For each $\Lambda$ there is a natural mapping $\tilde{m}_{\Lambda}:{\cal
A}_{\Lambda}\rightarrow{\cal A}^{(0)}$ corresponding to the homomorphism
$m_{\Lambda}:\Lambda\rightarrow{\bf R}$ that evaluates a numerical part of an
element from $\Lambda$. Then, ${\cal B}_{\Lambda}$ consists of those elements
of ${\cal A}_{\Lambda}$ that project onto ${\cal B}^{(0)}$ under
$\tilde{m}_{\Lambda}$. The mapping $\tilde{\alpha}:{\cal
B}_{\Lambda}\rightarrow{\cal B}_{\Lambda^{\prime}}$ assigned to a
homomorphisms $\alpha:\Lambda\rightarrow\Lambda^{\prime}$ is a restriction of
the corresponding map defined for the supermanifold $\cal A$. Let us call the
supermanifold $\cal B$ obtained this way the restriction of supermanifold
$\cal A$ to the subset ${\cal B}_{0}$ of its body.

One can define classes of supermanifolds with interesting geometric properties
restricting the allowed class of coordinate transformations. For example one
can define a complex analytic transformation of a $(2k\oplus
2l|2q)$-dimensional superdomain generalizing the usual requirement that the
Jacobian matrix of transformation commutes with a standard matrix $J$ obeying
$J^{2}=-1$. Then, a $(k\oplus l|q)$-dimensional complex manifold can be
defined as a manifold glued together by complex analytic transformations. One
can also use complex coordinates
$(Z^{A})=(z^{1},\dots,z^{k},w^{1},\dots,w^{l},\vartheta^{1},\dots,\vartheta^{q})$
on a complex manifold. They take values in complex AN algebras with antilinear
involution. Of course together with these coordinates we should consider
complex conjugate coordinates $\bar{Z}^{A}$. Analytic transformations do not
mix $Z^{A}$ and $\bar{Z}^{A}$. Therefore usually we will not mention
$\bar{Z}^{A}$.

Let $U$ be a $(1|N)$-dimensional complex superdomain with complex coordinates
$(z,\theta^{1},\dots,\theta^{N})$. We define N-superconformal transformations
as complex analytic transformations preserving up to a factor the one-form
$\omega_{N}=dz+\theta^{1}d\theta^{1}+\dots+\theta^{N}d\theta^{N}$ (up to a
factor means here up to multiplication by a nonvanishing superfield). For
example in case $N=1$ superconformal transformations can be written explicitly
as follows

$\displaystyle\tilde{z}$ $\displaystyle=$ $\displaystyle
u(z)-u^{\prime}(z)\epsilon(z)\theta$ $\displaystyle\tilde{\theta}$
$\displaystyle=$
$\displaystyle\sqrt{u^{\prime}(z)}\left(\theta+\epsilon(z)+\frac{1}{2}\epsilon(z)\epsilon^{\prime}(z)\theta\right)$
(4)

where $u(z)$ and $\epsilon(z)$ are even and odd analytic functions of $z$
respectively . By a N-superconformal manifold we mean a manifold pasted
together from $(1|N)$-dimensional complex superdomains by N-superconformal
transformations. Note that strictly speaking we are looking here not at a
single supermanifold but rather at a family of supermanifolds parameterized by
gluing functions similar to $u(z)$ and $\epsilon(z)$ in $N=1$ case. The
superspace $M_{N}$ of equivalence classes of N-superconformal manifolds is
called a moduli space of N-superconformal structures (supermoduli space).

An N-superconformal vector field defined in a complex superdomain $U^{1|N}$ is
a vector field $X$ such that the Lie derivative of the form $\omega_{N}$
restricted to $U^{1|N}$ with respect to $X$ is proportional to $\omega_{N}$,
i.e. $L_{X}\omega_{N}=f\omega_{N}$ for some superfield $f$. Such vector fields
correspond to infinitesimal N-superconformal transformations. Given an
N-superconformal compact manifold $\cal A$ one can define an N-superconformal
vector field on it as a vector field such that its restriction to each
elementary coordinate patch $U^{1|N}$ is N-superconformal. It follows from
standard results of deformation theory that a formal tangent space to the
moduli space of N-superconformal structures at a “point” $\cal A$ (being an
N-superconformal manifold) is isomorphic to $H^{1}({\cal A},\gamma_{N})$. Here
$H^{1}({\cal A},\gamma_{N})$ stands for the first cohomology group of $\cal A$
with coefficients in the sheaf of N-superconformal vector fields $\gamma_{N}$.

Moduli spaces play the central role in the Segal’s axiomatics of conformal
field theory (CFT) ( [6]). In this approach one considers conformal 2d
surfaces (complex curves) with parametrized boundary components. Each boundary
component is homeomorphic to a circle and it is assumed that the
parametrization can be extended to a complex coordinate in a small
neighborhood. We can think of a standard annulus in the complex plane
$\\{z\in{\bf C}|\frac{1}{2}\leq|z|\leq 1\\}$ mapped by a biholomorphic mapping
into a neighborhood of the boundary component. The neighborhoods of different
boundary components are assumed to be nonoverlapping. The annuli are divided
into two classes: “incoming” and “outgoing”. The moduli space of such objects
with $m$ incoming and $n$ outgoing annuli is denoted by $P_{m,n}$. Let us
stress here that we allow disconnected surfaces as well. There arise naturally
two operations on the sets $P_{m,n}$:

$\displaystyle P_{m_{1},n_{1}}\times P_{m_{2},n_{2}}$
$\displaystyle\rightarrow$ $\displaystyle P_{m_{1}+n_{1},m_{2}+n_{2}}$ (5)
$\displaystyle P_{m,n}$ $\displaystyle\rightarrow$ $\displaystyle P_{m-1,n-1}$
(6)

Here the first operation corresponds to the disjoint union of surfaces and the
second one corresponds to the identification of $m$-th incoming annulus with
the $n$-th outgoing one by the rule: $z^{\prime}=\frac{1}{4}z^{-1}$. In
Segal’s axiomatics CFT is specified by maps

$\alpha_{m,n}:P_{m,n}\rightarrow Hom_{\bf C}\\{H^{m},H^{n}\\}$

assigning to each point in $P_{m,n}$ a linear mapping $H^{m}\rightarrow H^{n}$
belonging to the trace class (here $H$ is a fixed Hilbert space). The
collection of mappings $\alpha_{m,n}$ should satisfy some set of axioms
ensuring the compatibility of mappings $\alpha_{m,n}$ with mappings (5), (6)
and with permutations of boundary components. One can generalize Segal’s
axiomatics to the case of N-superconformal field theories replacing 2d
conformal surfaces by N-superconformal manifolds with boundaries and complex
annuli by N-superconformal annuli (see [5] for this type of axiomatics stated
for N=2 SCFT).

Now let us describe supermanifolds that appear in 2D superconformal field
theories (SCFT) having different number of supersymmetries for left movers and
right movers. Consider a $(2|p+q)$-dimensional complex domain $U$ with even
coordinates $z_{L},z_{R}$ and odd coordinates
$\theta_{L}^{1},\dots,\theta_{L}^{p},\theta_{R}^{1},\dots,\theta_{R}^{q}$. The
superdomain $U$ can be considered as a superdomain in real superspace ${\bf
R}^{4|2p+2q}$. We will single out a subspace $V$ of $U$ that has real
dimension $(2\oplus 2|2p+2q)$ by imposing the condition that
$z_{R}-\bar{z}_{L}$ is nilpotent. By definition a transformation of $V$ is
called $(p,q)$-superconformal if it does not mix the left coordinates
$z_{L},\theta_{L}^{i}$ with the right coordinates $z_{R},\theta_{R}^{i}$ and
preserves up to a factor one-forms

$\displaystyle\omega_{L}$ $\displaystyle=$ $\displaystyle
dz_{L}+\theta_{L}^{1}d\theta_{L}^{1}+\dots+\theta^{p}_{L}d\theta_{L}^{p}$
$\displaystyle\omega_{R}$ $\displaystyle=$ $\displaystyle
dz_{R}+\theta_{R}^{1}d\theta_{R}^{1}+\dots+\theta^{q}_{R}d\theta_{R}^{q}$

We define a $(p,q)$-superconformal manifold as a superspace pasted together
from several copies of $V$ by means of $(p,q)$-superconformal transformations.
Again we assume here that the odd gluing parameters such as $\epsilon(z)$ in
(4) are allowed.

Let ${\cal A}^{(0)}$ be a compact connected oriented 2-dimensional surface of
genus $g>1$. It is easy to define a moduli space ${\cal M}_{p,q,g}$ of
$(p,q)$-superconformal manifolds having ${\cal A}^{(0)}$ as a body. Denote by
${\cal M}_{p,q,g}^{(0)}$ the body of moduli space ${\cal M}_{p,q,g}$. One can
prove that the superspace ${\cal M}_{p,q,g}$ can be constructed out of
supermoduli spaces ${\cal M}_{p,g}$ and ${\cal M}_{q,g}$ of $p$ and
$q$-superconformal manifolds with body ${\cal A}^{(0)}$. The construction goes
as follows. As in the case of moduli space of conformal structures ${\cal
M}_{p,g}$ is equipped with a canonical complex structure. Thus we may consider
the superspace ${\cal M}_{p,g}\times\bar{\cal M}_{q,g}$ where $\bar{\cal
M}_{q,g}$ denotes the space ${\cal M}_{q,g}$ with conjugate complex structure.
Note that there is a natural projection $\pi_{p}:{\cal
M}_{p,g}\rightarrow{\cal M}_{g}$ to the moduli space of complex structures on
${\cal A}^{(0)}$. This projection simply corresponds to the fact that each
p-superconformal manifold has a complex structure on its body. If $z$ is a
complex coordinate on ${\cal M}_{g}$ and $\bar{z}$ is its counterpart on
$\bar{\cal M}_{g}$ then we define the subset ${\cal
M}_{p,q,g}^{(0)}\subset{\cal M}_{p,g}^{(0)}\times\bar{\cal M}_{q,g}^{(0)}$ as
the set of points $(a,b)$ such that $\pi_{p}(a)=(\pi_{q}(b))^{*}$. Here $*$
stands for the conjugated complex structure. The superspace ${\cal M}_{p,q,g}$
has the body ${\cal M}_{p,q,g}^{(0)}$ and can be obtained as a restriction of
${\cal M}_{p,g}\times\bar{\cal M}_{q,g}$ to the corresponding subset of its
body. Choosing a local coordinate system on ${\cal M}_{p,g}\times\bar{\cal
M}_{q,g}$ in such a way that $z,\bar{z}$ are part of the coordinates we see
that the condition above implies that $z-(\bar{z})^{*}$ can take only
nilpotent values (as this is zero on the body). This means that if ${\cal
M}_{p,g}$ is of dimension $(k|l)$ and ${\cal M}_{q,g}$ is of dimension
$(\tilde{k}|\tilde{l})$ the superspace ${\cal M}_{p,q,g}$ has the dimension
$((k+\tilde{k}-3g+3)\oplus(3g-3)|l+\tilde{l})$. The construction of ${\cal
M}_{p,q,g}$ presented above is simply a more formal way to say that left and
right superconformal structures on a $(p,q)$-superconformal manifold are
independent up to a complex structure on the body that they share. It is easy
to generalize this construction to the case when ${\cal A}_{0}$ is
disconnected.

Now we are in a position to generalize Segal’s axiomatics to include
$(p,q)$-superconformal field theories. To define a proper analog of the space
$P_{m,n}$ one should consider a larger moduli space of (not necessarily
connected) $(p,q)$-superconformal surfaces having parameterized boundary
components of different type (determined by the boundary conditions for odd
coordinates). Furthermore, the corresponding analogs of the mappings
$\alpha_{m,n}$ should be holomorphic. The last requirement sets a connection
between $p$-superconformal and $(p,p)$-superconformal theories as outlined
below. Firstly, it is possible to embed the supermoduli space ${\cal M}_{p,g}$
into ${\cal M}_{p,p,g}$. Then , given a real analytic function on ${\cal
M}_{p,g}\subset{\cal M}_{p,p,g}$ one can extend it to a holomorphic function
defined in some neighborhood of ${\cal M}_{p,g}$ in ${\cal M}_{p,p,g}$.
Assuming that a continuation to the whole ${\cal M}_{p,p,g}$ is possible we
see that a $p$-superconformal field theory determines a $(p,p)$-superconformal
one.

Now let us make some remarks about the integration over ${\cal M}_{p,q,g}$.
This question is important in particular in heterotic string theory. To
calculate a string amplitude one defines a holomorphic volume element on
${\cal M}_{p,q,g}\times\bar{\cal M}_{p,q,g}$ and chooses a real cycle of
integration on the body of this space. We would like to stress that possible
reality conditions for odd and nilpotent variables do not affect the
integration result (see [7] or [8] for discussion ). Simply an integration
over odd variables is essentially an algebraic operation . As for the choice
of real cycle on the body of ${\cal M}_{p,q,g}\times\bar{\cal M}_{p,q,g}$ one
can take the (real) diagonal of the body of ${\cal M}_{p,q,g}\times\bar{\cal
M}_{p,q,g}$. The change of nilpotent variable roughly speaking corresponds to
the infinitesimal change of integration cycle and therefore does not change
the value of integral.

## Appendix

In this appendix we develop the approach to definition of superspace in terms
of functors (see [3]).

The notion of superspace.  Let us introduce a notion of almost nilpotent
algebra (or AN algebra ). An AN algebra is an associative finite dimensional
$Z_{2}$-graded supercommutative algebra $A$ with unit element such that its
ideal of nilpotent elements has codimension 1. In other words $A$ can be
decomposed (as a vector space) in a direct sum $A=R+N$ where $R$ is the
canonically embedded ground field ( R or C ) and $N=N(A)$ consists of
nilpotent elements. An example of AN algebra is a Grassmann algebra. Another
important example is an algebra generated by unit element $1$ and an element
$x$ satisfying the only relation $x^{n}=0$. An example including the previous
two is a supercommutative algebra with odd generators
$\xi_{1},\xi_{2},\cdots,\xi_{n}$ and even generators
$x_{1},x_{2},\cdots,x_{m}$ satisfying the relations
$x_{1}^{n_{1}}=0,x_{2}^{n_{2}}=0,\cdots,x_{m}^{n_{m}}=0$. Below parity
preserving homomorphisms of AN algebras are called morphisms .

Now we define a superspace $S$ as a rule assigning to each AN algebra
$\Lambda$ a set $S_{\Lambda}$ that we call the set of $\Lambda$-points of $S$
and to each morphism of two AN algebras
$\alpha:\Lambda\rightarrow\Lambda^{\prime}$ a map
$\tilde{\alpha}:S_{\Lambda}\rightarrow S_{\Lambda^{\prime}}$ of the
corresponding sets in a way that is consistent with compositions of morphisms.
The last assertion means that the map $\widetilde{\alpha_{2}\alpha_{1}}$
corresponding to the composition of morphisms $\alpha_{2}\alpha_{1}$ is equal
to the composition of maps $\tilde{\alpha}_{2}\tilde{\alpha}_{1}$. We consider
all AN algebras together with parity preserving homomorphisms between them as
a category $\bf AN$. In standard mathematical terminology superspace is a
covariant functor from the category $\bf AN$ to the category of sets. The
space $S_{R}$ corresponding to an AN algebra R (i.e. an AN algebra having a
trivial nilpotent part N) is called the body of superspace $S$. For every AN
algebra $\Lambda$ there is a homomorphism $m:\Lambda\rightarrow R$ assigning
to an element of $\Lambda$ its projection onto R with respect to the canonical
decomposition $\Lambda={R+N(\Lambda)}$. The corresponding map
$\tilde{m}:S_{\Lambda}\rightarrow S_{R}$ associates with each $\Lambda$-point
$a\in S_{\Lambda}$ a point $\tilde{m}(a)$ in the body of $S$ that we call the
numeric part of $a$. From now on if not specified we will assume our ground
field is the field of real numbers $\bf R$. Here we give some basic examples
of superspaces.

Example 1 Let $V$ be a $Z_{2}$ graded vector space, i.e. $V=V_{0}\oplus V_{1}$
where $V_{0}$ and $V_{1}$ are called the even and odd subspaces respectively.
We define the set $V_{\Lambda}$ to be the set of all formal (finite) linear
combinations $\sum_{i}e_{i}\alpha_{i}+\sum_{j}f_{j}\beta_{j}$ where $e_{i}\in
V_{0},f_{j}\in V_{1}$ , $\alpha_{i}$ and $\beta_{j}$ are respectively
arbitrary even and odd elements from $\Lambda$ (we assume the following
identifications $(\alpha_{1}+\alpha_{2})v=\alpha_{1}v+\alpha_{2}v$ ,
$\alpha(v_{1}+v_{2})=\alpha v_{1}+\alpha v_{2}$ where
$\alpha,\alpha_{1},\alpha_{2}\in\Lambda$ and $v,v_{1},v_{2}\in V$ ). The
mapping $\tilde{\rho}$ corresponding to a morphism
$\rho:\Lambda\rightarrow\Lambda^{\prime}$ carries the point
$\sum_{i}e_{i}\alpha_{i}+\sum_{j}f_{j}\beta_{j}$ to
$\sum_{i}e_{i}\rho(\alpha_{i})+\sum_{j}f_{j}\rho(\beta_{j})$ that obviously
lies in $V_{\Lambda^{\prime}}$. One can easily see that the maps
$\tilde{\rho}$ behave properly under compositions. Note that each set
$V_{\Lambda}$ is a linear space itself and is equipped with a canonical
structure of (left or right) $\Lambda_{0}$ module where $\Lambda_{0}$ is an
even part of $\Lambda$. For the last statement we set
$\mu(\sum_{i}e_{i}\alpha_{i}+\sum_{j}f_{j}\beta_{j})=\sum_{i}e_{i}(\mu\alpha_{i})+\sum_{j}f_{j}(\mu\beta_{j})$
for any $\mu\in\Lambda_{0}$ (this is for the left $\Lambda_{0}$ module
structure, the modification for the structure of right $\Lambda_{0}$ module is
obvious). We call superspaces with this property linear superspaces. The
construction given above works for any $Z_{2}$-graded linear space, including
infinite-dimensional ones. In the finite-dimensional case the corresponding
superspace is called $(p|q)$-dimensional linear superspace and is denoted by
${\bf R}^{p|q}$ (here $p=dimV_{0},q=dimV_{1}$). Choosing a basis in $V_{0}$
and $V_{1}$ one obtains a representation of ${\bf R}^{p|q}_{\Lambda}$ as a set
of $(p+q)$-tuples of elements from $\Lambda$ where first $p$ elements are from
$\Lambda_{0}$ and the next $q$ elements are from $\Lambda_{1}$. It readily
follows that the body of ${\bf R}^{p|q}$ is naturally isomorphic (as a linear
space) to ${\bf R}^{p}$.

Example 2 Here we want to describe some general operation over superspaces.
Let $\cal M$ be a superspace with a body ${\cal M}^{(0)}\equiv{\cal M}_{R}$.
Let ${\cal N}^{(0)}\subset{\cal M}^{(0)}$ be any subset. Define ${\cal
N}_{\Lambda}=(\tilde{m}({\cal N}^{(0)}))^{-1}\subset{\cal M}_{\Lambda}$ where
$\tilde{m}$ is the described above map evaluating the numerical part of
$\Lambda$-point. For any morphism of AN algebras
$\rho:\Lambda\rightarrow\Lambda^{\prime}$ we can define a map ${\cal
N}_{\Lambda}\rightarrow{\cal N}_{\Lambda^{\prime}}$ as the restriction of
$\tilde{\rho}$ to ${\cal N}_{\Lambda}$. Then it is easy to check that the sets
${\cal N}_{\Lambda}$ together with maps $\tilde{\rho}$ form a superspace. This
new superspace has the set ${\cal N}^{(0)}$ as its body. The superspace
constructed this way out of the given superspace $\cal M$ and a subset ${\cal
N}^{(0)}\subset{\cal M}^{(0)}$ will be called the restriction of superspace
$\cal M$ to the set ${\cal N}^{(0)}$. Given a domain ${\rm U}\subset{{\bf
R}^{p}}$ we can construct a restriction of superspace ${\bf R}^{p|q}$ to $\rm
U$ which is called a $(p|q)$-dimensional superdomain and denoted as $\rm
U^{p|q}$.

Example 3 Given a linear subspace ${\bf R}^{m}\subset{\bf R}^{p}$ one can
construct the restriction of superspace ${\bf R}^{p|q}$ to this subset. We
denote this new superspace by ${\bf R}^{m\oplus n|q}$ where $n=p-m$. The
explicit construction of it is as follows. Let $V=V_{0}\oplus V_{1}$ be a
$Z_{2}$ graded vector space and $dimV_{0}=p,dimV_{1}=s$. Given a decomposition
$V_{0}=U_{0}\oplus W_{0}$ ($dimU_{0}=m,dimW_{0}=n$) we define ${\bf
R}^{m\oplus n|s}_{\Lambda}$ to be the set of linear combinations of the form
$\sum_{i}d_{i}\alpha_{i}+\sum_{j}e_{j}\beta_{j}+\sum_{k}f_{k}\gamma_{k}$ where
$d_{i}\in U_{0},e_{j}\in W_{0},f_{k}\in V_{1}$ and
$\alpha_{i}\in\Lambda_{0},\beta_{j}\in\Lambda_{0}\cap
N(\Lambda),\gamma_{k}\in\Lambda_{1}$ (as above we require the natural
distributivity conditions). Given bases of $U_{0},W_{0},V_{1}$ we can
represent a point from ${\bf R}^{m\oplus n|s}_{\Lambda}$ as a $p+q$-tuple of
elements from $\Lambda$ in which first m elements are even, next n elements
are even and nilpotent and the last s elements are odd. Similarly to the
construction of ${\bf R}^{p|q}$ for any morphism of AN algebras $\rho$ we can
construct a mapping $\tilde{\rho}$ that carries the point
$\sum_{i}d_{i}\alpha_{i}+\sum_{j}e_{j}\beta_{j}+\sum_{k}f_{k}\gamma_{k}$ to
$\sum_{i}d_{i}\rho(\alpha_{i})+\sum_{j}e_{j}\rho(\beta_{j})+\sum_{k}f_{k}\rho(\gamma_{k})$.
Given a domain $\rm U\subset R^{m}$ the restriction of ${\bf R}^{m\oplus n|s}$
to $\rm U$ is called an $(m\oplus n|s)$-dimensional superdomain and is denoted
by $\rm U^{m\oplus n|s}$. Note that like ${\bf R}^{p|q}$, ${\bf R}^{m\oplus
n|q}$ is a vector superspace.

In a more interesting situation we have a superspace $\cal M$ such that the
sets ${\cal M}_{\Lambda}$ are equipped with some additional structure and the
maps $\tilde{\rho}$ are consistent with it. In other words we can consider a
functor from the category $\bf AN$ to a category of sets with additional
structure , e.g. smooth manifolds, groups, Lie groups, Lie algebras, etc. In
the previous section we gave an example of this sort of object, namely a
vector superspace is a functor with values in linear spaces. Similarly, one
can define a topological superspace as a functor from the category AN to the
category of topological spaces. We define a smooth superspace $\cal M$ as a
functor with values in smooth manifolds satisfying some additional conditions
to be described in a moment. Recall that in the examples described above
spaces of $\Lambda$-points of linear superspaces were equipped with a
structure of $\Lambda_{0}$-module. Thus we will require that a tangent space
at each point in ${\cal M}_{\Lambda}$ is equipped with a structure of (left)
$\Lambda_{0}$-module. A mapping $\tilde{\alpha}:M_{\Lambda}\rightarrow
M_{\Lambda^{\prime}}$ corresponding to a morphism $\alpha$ of two AN algebras
is called $\Lambda_{0}$-smooth if the corresponding tangent map
$(\tilde{\alpha})_{*}$ commutes with $\Lambda_{0}$-module structures, i.e.
$(\tilde{\alpha})_{*}\circ\lambda_{0}=\alpha(\lambda_{0})\circ(\tilde{\alpha})_{*}$.

Next we introduce the notions of associative superalgebra, Lie superalgebra
and supergroup. An associative superalgebra is a functor $\cal A$ taking
values in the category of associative algebras. In addition we require that
each algebra ${\cal A}(\Lambda)$ is equipped with a structure of
$\Lambda_{0}$-module in a way that is consistent with associative
multiplication and the mappings $\tilde{\alpha}$. An example of an associative
superalgebra can be obtained in the following way. Given a $Z_{2}$-graded
associative algebra $A$ one can consider linear combinations whose
coefficients are elements of AN algebra $\Lambda$ with appropriate parity (as
in the construction of vector superspace). Setting
$(a\lambda)(b\mu)=\pm(ab)(\lambda\mu)$ for all $(a,b)\in
A,\quad\lambda,\mu\in\Lambda$ (where minus sign occurs only if $\lambda$ and
$b$ are both odd) we get an associative algebra associated with every $A$ and
AN algebra $\Lambda$. These associative algebras can be considered as algebras
on sets of $\Lambda$-points of a certain superspace. A Lie superalgebra is a
functor with values in Lie algebras equipped with a natural structure of
linear superspace. Analogously to the previous example one can construct a Lie
superalgebra out of any $Z_{2}$-graded Lie algebra. Another example is a Lie
superalgebra constructed out of an associative superalgebra. One simply
defines a Lie algebra structure on the sets of $\Lambda$-points via the
commutator.

A supergroup is a functor with values in the category of groups. An important
example is a supergroup $GL(m|n)$ whose set of $\Lambda$-points consists of
$(m+n)\times(m+n)$ nondegenerate block matrices with entries from $\Lambda$.
The entries of $m\times m$ and $n\times n$ blocks are even and the entries of
$m\times n$ and $n\times m$ ones are odd. A matrix is called nondegenerate if
the left inverse matrix exists (one can show that this implies the existence
of right inverse matrix ). The set of all such block matrices (not necessarily
nondegenerate) is an example of associative superalgebra.

We say that a supergroup $G$ acts on a superspace $\cal S$ if for any AN
algebra $\Lambda$ the group $G_{\Lambda}$ acts on the set ${\cal S}_{\Lambda}$
in a way that is consistent with mappings $\tilde{\alpha}$ between
$\Lambda$-points. Namely, if $\phi_{g}$ is the mapping of the set ${\cal
S}_{\Lambda}$ to itself corresponding to the element $g\in G_{\Lambda}$, then
$\tilde{\alpha}\circ\phi_{g}=\phi_{\tilde{\alpha}(g)}\circ\tilde{\alpha}$. The
collection of quotient spaces ${\cal S}_{\Lambda}/G_{\Lambda}$ is endowed with
the structure of superspace in a natural way.

Supermanifolds. In this section we would like to give a definition of a
supermanifold as a superspace with values in the category of smooth manifolds
which is glued together from $(m\oplus n|s)$-dimensional superdomains. To this
end we need to describe an appropriate class of gluing transformations.

We start with the definition of a mapping of one superspace into another. Let
$\cal N$ and $\cal M$ be superspaces. We say that there is a mapping $F:{\cal
N}\rightarrow{\cal M}$ if for every AN algebra $\Lambda$ there is a map
$F_{\Lambda}:{\cal N}_{\Lambda}\rightarrow{\cal M}_{\Lambda}$ and the maps
$F_{\Lambda}$ are consistent with maps $\tilde{\alpha}$ corresponding to the
morphisms of AN algebras $\alpha:\Lambda\rightarrow\Lambda^{\prime}$. In other
words diagrams of the following type are commutative

$\displaystyle{\cal N}_{\Lambda}$
$\displaystyle\smash{\mathop{\longrightarrow}\limits^{F_{\Lambda}}}$
$\displaystyle{\cal M}_{\Lambda}$
$\displaystyle\tilde{\alpha}\Big{\downarrow}$
$\displaystyle\Big{\downarrow}\tilde{\alpha}$ (7) $\displaystyle{\cal
N}_{\Lambda^{\prime}}$
$\displaystyle\smash{\mathop{\longrightarrow}\limits^{F_{\Lambda^{\prime}}}}$
$\displaystyle{\cal M}_{\Lambda^{\prime}}$

In standard terminology a mapping of superspaces is a natural transformation
of functors. Our next goal will be to describe all possible mappings $F:{\bf
R}^{p\oplus q|s}\rightarrow{\bf R}^{p^{\prime}\oplus q^{\prime}|s^{\prime}}$.
Note that for now we do not require our maps to be consistent with natural
$\Lambda_{0}$-module structure defined on ${\bf R}^{p\oplus q|s}_{\Lambda}$
and ${\bf R}^{p^{\prime}\oplus q^{\prime}|s^{\prime}}_{\Lambda}$, nor do we
impose any smoothness conditions on $F_{\Lambda}$. We introduce an algebra
$A_{p,q,s}$ consisting of the expressions of the form

$\omega=\sum_{i=0}^{s}\sum_{1\leq\alpha_{1}<\cdots<\alpha_{i}\leq
s}f^{\alpha_{1},\cdots,\alpha_{i}}\xi_{\alpha_{1}}\dots\xi_{\alpha_{i}}$ (8)

where

$f^{\alpha_{1},\cdots,\alpha_{i}}=\sum_{(\beta_{1},\cdots\beta_{p},\gamma_{1},\cdots,\gamma_{q})}G^{\alpha_{1},\cdots,\alpha_{i}}_{\beta_{1},\cdots\beta_{p},\gamma_{1},\cdots,\gamma_{q}}(m_{1},\cdots,m_{p})n_{1}^{\beta_{1}}\dots
n_{p}^{\beta_{p}}l_{1}^{\gamma_{1}}\dots l_{q}^{\gamma_{q}}$ (9)

are formal series in variables $n_{1},\cdots,n_{p},l_{1},\cdots,l_{q}$, the
coefficients of which are arbitrary functions of real variables
$m_{1},\cdots,m_{p}$. If one assumes that $\xi_{1},\cdots,\xi_{s}$ in (8) are
generators of Grassmann algebra one can define the parity of expression (8) in
a natural way. Given an even (odd) element $\omega\in A_{p,q,s}$ one can
construct a mapping $\alpha_{\Lambda}^{\omega}$ of ${\bf R}^{p+q|s}_{\Lambda}$
into ${\bf R}^{1|0}_{\Lambda}$ (respectively, into ${\bf R}^{0|1}_{\Lambda}$)
in the following way. Given a $\Lambda$-point
$X=(x_{1},\dots,x_{p},y_{1},\dots,y_{q},\theta_{1},\dots,\theta_{s})\in{{\bf
R}^{p+q|s}_{\Lambda}}$ to evaluate $\alpha_{\Lambda}^{w}(X)$ one should
substitute in (8), (9) odd elements $\theta_{i}$ for $\xi_{i}$ , numeric part
$m(x_{i})$ for $m_{i}$ , nilpotent part $x_{i}-m(x_{i})$ for $n_{i}$ and
$y_{i}$ for $l_{i}$. Since $y_{i}$ and $x_{i}-m(x_{i})$ are nilpotent all
formal power series terminate. Given an even element $\omega\in A_{p,q,s}$
with the vanishing free term one can construct a map
$\alpha_{\Lambda}^{\omega}:{\bf R}^{p\oplus q|s}_{\Lambda}\rightarrow{\bf
R}^{0\oplus 1|0}$ using the same substitutions as those described above. It
can be readily verified that the mappings $\alpha_{\Lambda}^{\omega}$ for
different $\Lambda$ are consistent, i.e. the corresponing diagrams of the form
(7) are commutative, and thus define a mapping of superspace ${\bf R}^{p\oplus
q|s}$ into the superspace ${\bf R}^{1|0}$, ${\bf R}^{0|1}$ or ${\bf
R}^{0\oplus 1|0}$ if $\omega$ is even, odd or even and nilpotent respectively.
In the same fashion, given a row $L$ consisting of $p^{\prime}$ even,
$q^{\prime}$ even and nilpotent and $s^{\prime}$ odd elements from $A_{p,q,s}$
one can construct a map $\alpha^{L}_{\Lambda}:{\bf R}^{p\oplus
q|s}_{\Lambda}\rightarrow{\bf R}^{p^{\prime}\oplus
q^{\prime}|s^{\prime}}_{\Lambda}$. The maps $\alpha^{L}_{\Lambda}$ for
different $\Lambda$ are mutually consistent and determine a map $\alpha^{L}$
of the corresponding superspaces. In fact maps of the form $\alpha^{L}$
exhaust all maps between superspaces ${\bf R}^{p\oplus q|s}$ and ${\bf
R}^{p^{\prime}\oplus q^{\prime}|s^{\prime}}$. More precisely, we have

###### Theorem 1

The constructed correspondence of rows of $p^{\prime}$ even, $q^{\prime}$ even
and nilpotent and $s^{\prime}$ odd elements from $A_{p,q,s}$ to mappings ${\bf
R}^{p\oplus q|s}\rightarrow{\bf R}^{p^{\prime}\oplus q^{\prime}|s^{\prime}}$
is a bijection.

Proof. The injectivity follows directly from the construction of
correspondence. Thus we only need to show that the map is onto. Assume the
mappings $\alpha_{\Lambda}:{\bf R}^{p\oplus q|s}_{\Lambda}\rightarrow{\bf
R}^{p^{\prime}\oplus q^{\prime}|s^{\prime}}_{\Lambda}$ define a mapping of
corresponding superspaces. Let $\Lambda_{k,r,N}$ be an AN algebra generated by
the even generators $a_{1},\dots,a_{k}$ satisfying the relations
$a_{1}^{N}=a_{2}^{N}=\dots=a_{k}^{N}=0$ and odd generators
$\eta_{1},\dots,\eta_{r}$. We assume the only relations in $\Lambda_{k,s,N}$
are due to supercommutativity and those relations on even generators that are
written above. Let $\Lambda$ be any AN algebra and
$x=(m_{1}+n_{1},\dots,m_{p}+n_{p},l_{1},\dots,l_{q},\theta_{1},\dots,\theta_{s})$
be any point from ${\bf R}^{p+q|s}_{\Lambda}$ (here $m_{i}+n_{i}$ is the
decomposition into numerical and nilpotent parts respectively). Denote by
$\tau_{m,N}$ a point
$(m_{1}+a_{1},\dots,m_{p}+a_{p},a_{p+1},\dots,a_{p+q},\eta_{1},\dots,\eta_{s})\in{\bf
R}^{p\oplus q|s}_{\Lambda_{p+q,s,N}}$ (the lower script $m$ stands for the
collection $(m_{1},\dots,m_{p})$) . We claim that if N is sufficiently large
there exists a morphism $\rho:\Lambda_{p+q,s,N}\rightarrow\Lambda$ such that
$\tilde{\rho}(\tau_{m,N})=x$. Indeed, let $n(\Lambda)$ be the maximal
nilpotency degree of elements in $\Lambda$. Then, for $N\geq n(\Lambda)$ the
map

$a_{1}\mapsto n_{1},\dots,a_{p}\mapsto n_{p}$ $a_{p+1}\mapsto
l_{1},\dots,a_{p+q}\mapsto l_{q}$
$\eta_{1}\mapsto\theta_{1},\dots,\eta_{s}\mapsto\theta_{s}$

can be extended up to a parity preserving homomorphism
$\rho:\Lambda_{p+q,s,N}\rightarrow\Lambda$. One can easily see that for $\rho$
constructed this way we have $\tilde{\rho}(\tau_{m,N})=x$. By commutativity of
diagram (7) corresponding to the homomorphism $\rho$ we have
$\tilde{\rho}\alpha_{\Lambda_{p+q,s,N}}=\alpha_{\Lambda}\tilde{\rho}$.
Therefore
$\alpha_{\Lambda}(x)=\alpha_{\Lambda}\tilde{\rho}(\tau_{m,N})=\tilde{\rho}\alpha_{\Lambda_{p+q,s,N}}(\tau_{m,N})$
which means that the image of $x$ under the map $\alpha_{\Lambda}$ is uniquely
determined by the image of $\tau_{m}$ under the map
$\alpha_{\Lambda_{p+q,s,N}}$. Since $\Lambda$ is an arbitrary AN algebra we
see that the mapping of superspaces $\alpha$ is determined by the images of
points $\tau_{m,N}\in{\bf R}^{p\oplus q|s}_{\Lambda_{p+q,s,N}}$ for different
$N$. Since the nilpotent parts of the coordinates of $\tau_{m,N}$ are the
generators of $\Lambda_{p+q,s,N}$ the coordinates of the image of $\tau_{m,N}$
under $\alpha_{\Lambda_{p+q,s,N}}$ are clearly polynomials in them with
coefficients depending on $m_{1},\dots,m_{p}$ and hence define elements from
$A_{p,q,s}$. The last step is to show that the polynomials corresponding to
different $N$ are consistent in a sense that polynomials corresponding to
algebras with smaller $N$ are just initial pieces of those corresponding to
the algebras with larger $N$. This is due to the fact that there exists a
homomorphism $\rho:\Lambda_{p+q,s,M}\rightarrow\Lambda_{p+q,s,N}$ for $M\geq
N$ such that $\tilde{\rho}(\tau_{m,M})=\tau_{m,N}$. The homomorphism $\rho$ is
defined by the following maps

$\tilde{a}_{i}\mapsto a_{i},\quad\tilde{\eta}_{j}\mapsto\eta_{j}$

where $\tilde{a}_{i},a_{i}$ are even and $\tilde{\eta}_{j},\eta_{j}$ are odd
generators of $\Lambda_{p+q,s,M}$ and $\Lambda_{p+q,s,N}$ respectively. The
same use of commutative diagram (7) for thus constructed $\rho$ as above
assures that the polynomials corresponding to
$\alpha_{\Lambda_{p+q,s,N}}(\tau_{m})$ are consistent and thus define in a
unique way a row of $p^{\prime}$ even, $q^{\prime}$ even and nilpotent and
$s^{\prime}$ odd elements from $A_{p,q,s}$. $\Box$

Let us turn now to some additional requirements that we impose on admissible
mappings of supermanifolds. First note that if $\Lambda$ is an AN algebra such
that $dim\Lambda_{0}=k,dim\Lambda_{1}=l$ then we have an isomorphism ${\bf
R}^{p\oplus q|s}_{\Lambda}\cong{\bf R}^{L},\quad L=pk+q(k-1)+sl$. Since the
tangent space $\rm TR^{L}$ is isomorphic to ${\bf R}^{L}$ itself, for every
$\Lambda$ point $x\in{\bf R}^{p\oplus q|s}_{\Lambda}$ we have a $\Lambda_{0}$
module structure induced on $T_{x}$ in a natural way. We say that a smooth
mapping $\alpha_{\Lambda}:{\bf R}^{p\oplus q|s}_{\Lambda}\rightarrow{\bf
R}^{p^{\prime}\oplus q^{\prime}|s^{\prime}}_{\Lambda}$ is $\Lambda_{0}$-smooth
if for every $x\in{\bf R}^{p\oplus q|s}_{\Lambda}$ the tangent map
$(\alpha_{\Lambda})_{*}:T_{x}\rightarrow T_{\alpha_{\Lambda}(x)}$ is a
homomorphism of $\Lambda_{0}$ modules. The mapping $\alpha:{\bf R}^{p\oplus
q|s}\rightarrow{\bf R}^{p^{\prime}\oplus q^{\prime}|s^{\prime}}$ is said to be
smooth if for all AN algebras $\Lambda$ the maps $\alpha_{\Lambda}$ are
$\Lambda_{0}$-smooth. The description of smooth mappings between the spaces
${\bf R}^{p\oplus q|s}$ is given by theorem 2 which we will formulate in a
moment. But first we introduce the algebra $B_{p,q,s}$ as a tensor product of
$C^{\infty}({\bf R}^{p})$, an algebra of formal series with respect to $q$
variables $n_{1},\dots,n_{q}$ and a Grassmann algebra with $s$ generators
$\xi_{1},\dots,\xi_{s}$ :

$B_{p,q,s}=C^{\infty}({\bf R}^{p})\otimes{\bf
R}[[n_{1},\dots,n_{q}]]\otimes\Lambda^{.}(\xi_{1},\dots,\xi_{s})$

More explicitly, elements of $B_{p,q,s}$ are of the form (8) where
coefficients $f^{\alpha_{1},\cdots,\alpha_{i}}$ are series of the form

$f^{\alpha_{1},\cdots,\alpha_{i}}=\sum_{(\beta_{1},\cdots\beta_{q})}G^{\alpha_{1},\cdots,\alpha_{i}}_{\beta_{1},\cdots\beta_{q}}(x_{1},\cdots,x_{p})l_{1}^{\beta_{1}}\dots
l_{p}^{\beta_{q}}$ (10)

and
$G^{\alpha_{1},\cdots,\alpha_{i}}_{\beta_{1},\cdots\beta_{q}}(x_{1},\cdots,x_{p})$
are smooth functions of variables $x_{1},\cdots,x_{p}$. Again the $Z_{2}$
grading in the exterior algebra $\Lambda^{.}(\xi_{1},\dots,\xi_{s})$ induces a
$Z_{2}$ grading on $B_{p,q,s}$. One can check that given a $\Lambda$-point
from ${\bf R}^{p\oplus q|s}$ and an even (odd) element from $B_{p,q,s}$ by
substituting first $p$ coordinates instead of variables $x_{1},\dots,x_{p}$ ,
next $q$ coordinates instead of variables $l_{1},\dots,l_{q}$ and the last $s$
variables instead of $\xi_{1},\dots,\xi_{s}$ one gets an even (odd) element
from $\Lambda$. These substitutions make sense because one can use the Taylor
expansion of functions $G$ in the nilpotent parts of the first $p$ variables
and the formal series in the next $q$ variables terminate due to nilpotency.
In parallel with the case of algebra $A_{p,q,s}$ one can check that a row of
$p^{\prime}+q^{\prime}+s^{\prime}$ elements from $B_{p,q,s}$ with appropriate
parities and nilpotency properties defines a mapping ${\bf R}^{p\oplus
q|s}\rightarrow{\bf R}^{p^{\prime}\oplus q^{\prime}|s^{\prime}}$. Moreover,
this map is smooth in the sense explained above. Note that such a row also
defines a parity preserving homomorphism
$B_{p^{\prime},q^{\prime},s^{\prime}}\rightarrow B_{p,q,s}$ (the generators
are sent to the corresponding elements of the row). In fact the following
theorem is valid:

###### Theorem 2

The smooth mappings ${\bf R}^{p\oplus q|s}\rightarrow{\bf R}^{p^{\prime}\oplus
q^{\prime}|s^{\prime}}$ are in a bijective correspondence with the parity
preserving homomorphisms $B_{p^{\prime},q^{\prime},s^{\prime}}\rightarrow
B_{p,q,s}$ .

Proof. We have shown how to construct a smooth mapping ${\bf R}^{p\oplus
q|s}\rightarrow{\bf R}^{p^{\prime}\oplus q^{\prime}|s^{\prime}}$ out of a row
of $p^{\prime}$ even, $q^{\prime}$ even and nilpotent and $s^{\prime}$ odd
elements from $B_{p,q,s}$. Conversely, given a mapping $\alpha:{\bf
R}^{p\oplus q|s}\rightarrow{\bf R}^{p^{\prime}\oplus q^{\prime}|s^{\prime}}$
by theorem 1 we can assign to it a row of elements from $A_{p,q,s}$. Let
$\omega_{i}=\omega_{i}(m,n,l,\xi)$ (see (8), (9) ) be the $i$-th coordinate of
this row. From the assumption of $\Lambda_{0}$-smoothness it follows directly
that the coefficients
$G^{\alpha_{1},\cdots,\alpha_{j}}_{\beta_{1},\cdots\beta_{p},\gamma_{1},\cdots,\gamma_{q}}(m_{1},\cdots,m_{p})$
corresponding to elements $\omega_{i}$ are smooth functions.

Let $\lambda=m(\lambda)+n(\lambda)$ be an element of $\Lambda_{0}$. Set
$m(\lambda)\epsilon$ and $n(\lambda)\epsilon$ to be an increment of numerical
and nilpotent parts of the $k$-th coordinate of the space ${\bf R}^{p\oplus
q|s}_{\Lambda}$ respectively ( $k\leq p$, $\epsilon$ is a small real number).
All other coordinates except the $k$-th one are fixed. Then, by
$\Lambda_{0}$-smoothness

$\omega_{i}(m_{k}+m(\lambda)\epsilon,n_{k}+n(\lambda)\epsilon)-\omega_{i}(m_{k},n_{k})=\lambda(\omega(m_{k}+\epsilon,n_{k})-\omega(m_{k},n_{k}))+o(\epsilon)$

On the other hand the LHS of the last equation can be written as

$m(\lambda)\epsilon\frac{\partial\omega_{i}(m_{k},n_{k})}{\partial
m_{k}}+n(\lambda)\epsilon\frac{\partial\omega_{i}(m_{k},n_{k})}{\partial
n_{k}}+o(\epsilon)$

from which we conclude that $\frac{\partial\omega_{i}(m_{k},n_{k})}{\partial
n_{k}}=\frac{\partial\omega_{i}(m_{k},n_{k})}{\partial m_{k}}$. The last
equation being written in terms of series
$\omega_{i}=b_{0}+b_{1}n_{k}+b_{2}n_{k}^{2}+\dots$ where $b_{i}$ depend on
$m,l,\xi$ and all $n$’s except $n_{k}$ reads as

$b_{1}+2b_{2}n_{k}+3b_{3}n_{k}^{2}+\dots=\frac{\partial b_{0}}{\partial
m_{k}}+\frac{\partial b_{1}}{\partial m_{k}}n_{k}+\frac{\partial
b_{2}}{\partial m_{k}}n_{k}^{2}+\dots$

Therefore,

$\omega_{i}=b_{0}+\frac{1}{1!}\frac{\partial b_{0}}{\partial
m_{k}}n_{k}+\frac{1}{2!}\frac{\partial^{2}b_{0}}{\partial
m_{k}^{2}}n_{k}^{2}+\dots$

Hence $\omega_{i}=b_{0}(m_{k}+n_{k})$ where for shortness all other arguments
are skipped. Repeating this argument successively for all $k=1,\dots,p$ we get

$\omega_{j}=\sum_{i=0}^{s}\sum_{1\leq\alpha_{1}<\cdots<\alpha_{i}\leq
s}f_{j}^{\alpha_{1},\cdots,\alpha_{i}}\xi_{\alpha_{1}}\dots\xi_{\alpha_{i}}$

where

$f_{j}^{\alpha_{1},\cdots,\alpha_{i}}=\sum_{(\gamma_{1},\cdots,\gamma_{q})}G^{\alpha_{1},\cdots,\alpha_{i}}_{j;0,\cdots
0,\gamma_{1},\cdots,\gamma_{q}}(m_{1}+n_{1},\cdots,m_{p}+n_{p})l_{1}^{\gamma_{1}}\dots
l_{q}^{\gamma_{q}}$

Compare with (8), (10). The last two expressions mean precisely that $\omega$
is given by a row of $p^{\prime}+q^{\prime}+s^{\prime}$ elements (with
corresponding parity and nilpotency properties) from the algebra $B_{p,q,s}$,
i.e. represents a parity preserving homomorphism
$B_{p^{\prime},q^{\prime},s^{\prime}}\rightarrow B_{p,q,s}$. $\Box$

The theorems 1 and 2 are modifications of statements proved in [4].

The theorem 2 has a straightforward generalization to the case of mappings
between superdomains. Let $B_{p,q,s}(U)={\bf R}[[n_{1},\dots,n_{q}]]\otimes
C^{\infty}({U})\otimes\Lambda^{.}(\xi_{1},\dots,\xi_{s})$. Then, we claim that
smooth mappings of superdomains $\rm U^{p+q,s}\rightarrow
V^{p^{\prime}+q^{\prime},s^{\prime}}$ are in a bijective correspondence with
parity preserving homomorphisms
$B_{p^{\prime},q^{\prime},s^{\prime}}(V)\rightarrow B_{p,q,s}(U)$ (one can
easily modify the proof taking into account the restriction on the range of
first $p$ (respectively $p^{\prime}$) coordinates). Now let us give a
definition of supermanifold.

###### Definition

A $(p\oplus q|s)$-dimensional supermanifold $S$ is a superspace which is glued
together from $(p\oplus q|s)$-dimensional superdomains by means of smooth
mappings.

In more detail, we assume that there exists a covering $\cal U$ of the body
$S_{\bf R}$ of superspace $S$ with the property that a restriction of $S$ to
any $U\in{\cal U}$ which we denote as $S^{U}$ is isomorphic to the superdomain
$U_{\alpha}^{p\oplus q|s}$. Let $U$ and $V$ be any two intersecting sets from
the covering $\cal U$ and let $\phi_{U}:S^{U}\rightarrow U_{\alpha}^{p\oplus
q|s},\phi_{V}:S^{V}\rightarrow V_{\alpha}^{p\oplus q|s}$ be the corresponding
isomorphisms. Denote by $\tilde{\phi}_{U},\tilde{\phi}_{V}$ the isomorphisms
$S^{U\cap V}\rightarrow(U\cap V)_{\alpha}^{p\oplus q|s}$ induced by $\phi_{U}$
and $\phi_{V}$ respectively. Then we require that the mappings
$(\tilde{\phi}_{U})^{-1}\tilde{\phi}_{V}$ are smooth.

Using theorem 2 one can give a definition of a $(p\oplus q|s)$-dimensional
supermanifold along the lines of a conventional Berezin-Leites approach to
supermanifolds via ringed spaces. In those terms, a $(p\oplus
q|s)$-dimensional supermanifold is a manifold ${\cal M}_{0}$ with a sheaf
$\cal O$ of supercommutative rings on it with the property that locally over a
neighborhood $U$ the sheaf is isomorphic to the algebra
$B_{p,q,s}(U)=C^{\infty}({U})\otimes{\bf
R}[[n_{1},\dots,n_{q}]]\otimes\Lambda^{.}(\xi_{1},\dots,\xi_{s})$.

On the other hand, given a supermanifold $\cal M$ one can construct a sheaf of
$Z_{2}$-graded supercommutative algebras over the body ${\cal M}_{0}$ as
follows. For any open subset $U\subset{\cal M}_{0}$ consider the space of
mappings from the restriction of $\cal M$ to $U$ into ${\bf R}^{1|1}$. Denote
this space by ${\cal F}(U)$. Note that one can identify ${\bf
R}^{1|1}_{\Lambda}$ with $\Lambda$. This induces a structure of $Z_{2}$-graded
supercommutative algebra on ${\cal F}(U)$ . It is easy to see that the
collection of ${\cal F}(U)$ is a (pre)sheaf endowing $\cal M$ with a structure
of ringed space. The fact that locally our supermanifold is isomorphic to a
superdomain determines the standard local structure of the sheaf at hand.

We define a tangent bundle $T{\cal M}$ of a supermanifold $M$ as a functor
determined by $\Lambda$-points $(T{\cal M})_{\Lambda}=T({\cal M}_{\Lambda})$
and mappings $\tilde{\alpha}_{*}$ corresponding to morphisms
$\alpha:\Lambda\rightarrow\Lambda^{\prime}$. One can easily check that $T{\cal
M}$ is a supermanifold itself and has a canonical projection $T{\cal
M}\rightarrow{\cal M}$.

It is possible to consider superspaces modelled on a fixed infinite-
dimensional topological linear superspace $V$. Such a superspace $V$ can be
constructed as in Example 2 out of any infinite-dimensional topological
$Z_{2}$-graded linear space. An infinite-dimensional superdomain is a
restriction of $V$ to any open subset of its body. In the infinite-dimensional
case one can also define objects analogous to $(k\oplus l|q)$-dimensional
linear superspaces. From now on let us fix $V$. One can give different
definitions of differentiable (smooth) mappings between two topological linear
spaces. For our purposes the following definition will be sufficient. We will
call a mapping $F:V_{1}\rightarrow V_{2}$ between two infinite-dimensional
topological linear superspaces differentiable if there exists a mapping
$D:V_{1}\times V_{1}\rightarrow V_{2}$ such that $D$ is linear in the second
variable and for each AN algebra $\Lambda$
$F_{\Lambda}(x+h)=F_{\Lambda}(x)+D_{\Lambda}(x,h)+o_{x}(h)$ for any
$x,h\in(V_{1})_{\Lambda}$. Here $o_{x}(h)$ vanishes to an order higher than
$h$, i.e. for a real number $t\to 0$ , $t^{-1}o_{x}(th)\to 0$. (this is what
is called differentiability in the sense of $\rm G{\hat{a}}teaux$ , see for
example [9]). Once we defined a differentiable mapping between superdomains in
$V$ we can consider infinite-dimensional differentiable supermanifolds
modelled on $V$ as superspaces pasted from superdomains in $V$ by means of
differentiable mappings.

Lie supergroups. We have already given above the definitions of supergroup and
supermanifold. Following the same category-theoretic point of view we define a
$(p\oplus q|s)$-dimensional Lie supergroup as a $(p\oplus q|s)$-dimensional
supermanifold which is also a supergroup and for each AN algebra $\Lambda$ the
set of $\Lambda$-points is a Lie group with respect to the given smooth
manifold and group structures. To any $(p\oplus q|s)$-dimensional Lie
supergroup one assigns naturally its Lie superalgebra being a $(p\oplus
q|s)$-dimensional vector space. All this can be generalized easily to include
infinite-dimensional Lie supergroups modelled on a linear superspace $V$.

Given any Lie superalgebra $\cal G$ (possibly infinite-dimensional) one can
consider its restriction to a point $0$ of the body and obtain a Lie
superalgebra ${\cal G}_{nilp}$ which has only nilpotent coordinates. Then, for
each AN algebra $\Lambda$ one can consider a set of formal power series
$G_{\Lambda}=\\{exp(a)|a\in({\cal G}_{nilp})_{\Lambda}\\}$. The set
$G_{\Lambda}$ is endowed with a Lie supergroup structure via the Campbell-
Hausdorf formula which is represented by a finite sum because of nilpotency.
It is easy to check that the sets $G_{\Lambda}$ constitute a Lie supergroup
$G_{nilp}$ modelled on a linear superspace $V={\cal G}_{nilp}$ and having
${\cal G}_{nilp}$ as its Lie superalgebra. If there exists a Lie supergroup
$G$ having $\cal G$ as its Lie superalgebra, then $G_{nilp}$ can alternatively
be constructed as the restriction of superspace $G$ to the subset of its body
$G^{(0)}$ consisting of one point \- the unit element of Lie group $G^{(0)}$.
These two constructions are canonically isomorphic ( the isomorphism is set up
via exponential mappings in $G_{\Lambda}$’s). In a more general situation it
might happen that ${\cal G}$ contains an invariant Lie subalgebra ${\cal
G}_{int}\subset{\cal G}$ such that there exists a Lie supergroup $G_{int}$
with ${\cal G}_{int}=Lie(G_{int})$ (the subscript “$int$” comes from the word
“integrable”). Then, the construction above can be improved as follows.

First note that the supergroup $G_{int}$ acts on $G_{nilp}$ by the adjoint
action on ${\cal G}_{nilp}$. Then we can consider a semidirect product of Lie
supergroups $G_{nilp}\times_{Ad}G_{int}$ , i.e. the direct product
supermanifold $G_{nilp}\times G_{int}$ with a group operation defined by the
following formula

$\left(e^{x},a\right)(e^{y},b)=(e^{x}(ae^{y}a^{-1}),ab)$

As it was explained $G_{int}$ contains $(G_{int})_{nilp}$ which is canonically
isomorphic to a Lie subgroup $(G_{int})_{nilp}\subset G_{nilp}$. It is
straightforward to check that the set of pairs $H=\\{(e^{x},e^{-x})\in
G_{nilp}\times G_{int}|e^{x}\in(G_{int})_{nilp}\\}$ is an invariant Lie
subgroup of $G_{nilp}\times_{Ad}G_{int}$. Taking the quotient with respect to
$H$ we get a Lie supergroup whose Lie superalgebra is isomorphic to the one
obtained as a restriction of Lie superalgebra $\cal G$ to a subset of its body
${\cal G}_{int}^{(0)}\subset{\cal G}^{(0)}$.

Superspaces of maps. In this section we introduce a superspace of maps and
give some explicit constructions of this object. Let $\cal E,F$ be
superspaces. For any superspace ${\cal S}$ a mapping $\sigma:{\cal
E}\times{\cal S}\rightarrow{\cal F}$ is called a family of mappings from the
superspace $\cal E$ into the superspace $\cal F$ with base $\cal S$. A family
of mappings $\nu:{\cal E}\times{\cal F^{E}}\rightarrow{\cal F}$ is called a
universal family of mappings ${\cal E}\rightarrow{\cal F}$ if it has the
following property . For any family ${\cal E}\times{\cal S}\rightarrow{\cal
F}$ there exists a unique mapping $\rho:{\cal S}\rightarrow{\cal F^{E}}$ such
that $\nu\circ(id_{\cal E}\times\rho)=\sigma$. In the case when $\cal E,F$ are
supermanifolds we assume that all mappings are smooth and the base $\cal S$ of
the family is a supermanifold though the base of the universal family need not
be a supermanifold. For any particular pair of superspaces $\cal E,F$ there
arises naturally a question of whether such a universal family of mappings
exists. Provided a universal family exists and is unique up to an isomorphism,
the base of the universal family is said to be the superspace of mappings
(between corresponding superspaces). Below we will construct superspaces of
maps between superdomains and describe a superspace of maps for the case of
supermanifolds.

Example. In this example we give a construction of the superspace of smooth
maps ${\bf R}^{p|q}\rightarrow{\bf R}^{p^{\prime}|q^{\prime}}$. Due to theorem
1 from section 3 the smooth mappings ${\bf R}^{p|q}\rightarrow{\bf
R}^{p^{\prime}|q^{\prime}}$ are in a bijective correspondence with parity
preserving homomorphisms $B_{p^{\prime},q^{\prime}}\rightarrow B_{p,q}$
($B_{p,q}\equiv B_{p,q,0}$). These homomorphisms form a vector space
$V^{p,q}_{p^{\prime},q^{\prime}}$ whose elements are rows of $p^{\prime}$ even
and $q^{\prime}$ odd elements from $B_{p,q}$. Likewise one can consider a
vector space $\tilde{V}^{p,q}_{p^{\prime},q^{\prime}}$ of parity reversing
homomorphisms . Its elements are rows of $p^{\prime}$ odd and $q^{\prime}$
even elements from $B_{p,q}$. Define a vector superspace ${\bf
V}^{p,q}_{p^{\prime},q^{\prime}}$ as a vector superspace associated with a
$Z_{2}$-graded vector space
$V^{p,q}_{p^{\prime},q^{\prime}}\oplus\tilde{V}^{p,q}_{p^{\prime},q^{\prime}}$.
We assert that ${\bf V}^{p,q}_{p^{\prime},q^{\prime}}$ can be considered as a
superspace of smooth mappings ${\bf R}^{p|q}\rightarrow{\bf
R}^{p^{\prime}|q^{\prime}}$. To an element $b\in{\bf
V}^{p,q}_{p^{\prime},q^{\prime}}(\Lambda)$ we assign a smooth mapping ${\bf
R}^{p|q}_{\Lambda}\rightarrow{\bf R}^{p^{\prime}|q^{\prime}}_{\Lambda}$ in a
natural way. One can check that this gives a family of smooth mappings
$\nu^{p,q}_{p^{\prime},q^{\prime}}:{\bf R}^{p|q}\times{\bf
V}^{p,q}_{p^{\prime},q^{\prime}}\rightarrow{\bf R}^{p^{\prime}|q^{\prime}}$.
Now let $\sigma:{{\bf R}^{p|q}}\times S\rightarrow{\bf
R}^{p^{\prime}|q^{\prime}}$ be a family of smooth mappings. The most important
case is $S={\bf R}^{m|n}$. Then, ${{\bf R}^{p|q}}\times S\cong{{\bf
R}^{p+m|q+n}}$ and by Theorem 2 , $\sigma$ corresponds to an element from
$V^{p+m,q+n}_{p^{\prime},q^{\prime}}$. This element can be considered as a row
of $p^{\prime}+q^{\prime}$ elements
$\sigma^{k},k=1,2,\dots,p^{\prime}+q^{\prime}$ from $B_{p+m,q+n}$ with the
corresponding parity. Explicitly

$\sigma^{k}=\sum_{i=0}^{q}\sum_{j=0}^{n}\sum_{\alpha,\beta}f^{k}_{\alpha_{1},\cdots,\alpha_{i},\beta_{1},\dots,\beta_{j}}(x^{1},\dots,x^{p},y^{1},\dots,y^{m})\xi^{\alpha_{1}}\dots\xi^{\alpha_{i}}\eta^{\beta_{1}}\dots\eta^{\beta_{j}}$

where we assume that variables $x^{i},\xi^{j}$ ( $y^{i},\xi^{i}$) are assigned
to the superspace ${\bf R}^{p|q}$ (respectively ${\bf R}^{m|n}$) and the
summation runs over the numbers $i$ and $j$ such that $i+j$ is of the
appropriate parity . The collection of elements from $B_{p,q}$

$\displaystyle g^{k}(x^{1},\dots,x^{p})_{\alpha_{1},\cdots,\alpha_{i}}$
$\displaystyle=$
$\displaystyle\sum_{j=0}^{n}\sum_{1\leq\beta_{1}<\cdots<\beta_{i}\leq
n}f^{k}_{\alpha_{1},\cdots,\alpha_{i},\beta_{1},\dots,\beta_{j}}(x^{1},\dots,x^{p},y^{1},\dots,y^{m})$
$\displaystyle\eta^{\beta_{1}}\dots\eta^{\beta_{j}}$

labeled by the sequence of multiindices $1\leq\alpha_{1}<\dots<\alpha_{i}\leq
q$, integer $k$, and the variables $x^{1},\dots,x^{p}$ defines a mapping
$\rho:{{\bf R}^{m|n}}\rightarrow{\bf V}^{p,q}_{p^{\prime},q^{\prime}}$. One
should simply utilize the standard substitutions of $\Lambda$-point
coordinates into $g^{k}(x^{1},\dots,x^{p})_{\alpha_{1},\cdots,\alpha_{i}}$ and
then form a combination

$\sum_{i=0}^{q}\sum_{1\leq\alpha_{1}<\cdots<\alpha_{i}\leq
q}\xi^{\alpha_{1}}\dots\xi^{\alpha_{i}}g^{k}(x^{1},\dots,x^{p})_{\alpha_{1},\cdots,\alpha_{i}}\in{\bf
V}^{p,q}_{p^{\prime},q^{\prime}}(\Lambda)$

A direct check shows that $\nu^{p,q}_{p^{\prime},q^{\prime}}\circ\rho=\sigma$.
Using the fact that locally every supermanifold is equivalent to a superdomain
one can prove that ${\bf V}^{p,q}_{p^{\prime},q^{\prime}}$ is indeed a
superspace of maps. Note that the same considerations as above go through for
the case of mappings $U^{p|q}\rightarrow{\bf R}^{p^{\prime}|q^{\prime}}$ of a
superdomain to a linear superspace. One simply needs to consider vector spaces
of parity preserving and parity reversing homomorphisms
$B_{p^{\prime},q^{\prime}}\rightarrow B_{p,q}(U)$ and repeat the construction
above. The superspace of maps between superdomains $U^{p|q}\rightarrow
V^{p^{\prime}|q^{\prime}}$ is the restriction of superspace of maps
$U^{p|q}\rightarrow{\bf R}^{p^{\prime}|q^{\prime}}$ to the subset of its body
consisting of maps carrying the body of the superdomain $U$ to the subset
$V\subset{\bf R}^{p^{\prime}|q^{\prime}}$. The generalization of these results
to the case of $(p\oplus q|s)$-dimensional superdomains is straightforward.
One should replace everywhere algebras of the type $B_{p,q}$ by the algebras
of the type $B_{p,q,s}$.

Now let us discuss briefly a situation in the case of mappings between two
supermanifolds. Let $\cal M$ and $\cal N$ be supermanifolds. Then, the
restriction of superspace of maps ${\cal N}^{\cal M}$ to each connected
component of its body can be endowed with a structure of infinite-dimensional
supermanifold. For a given map $\phi:{\cal M}\rightarrow{\cal N}$ consider the
pull-back of the tangent bundle $T{\cal N}$ induced by $\phi$. Denote this
bundle as $\phi^{*}T{\cal N}$. The superspace of sections of $\phi^{*}T{\cal
N}$ is a vector superspace which can be regarded as a tangent space to ${\cal
N}^{\cal M}$ at the point $\phi$ in a natural way. Putting some
(super)Riemannian metric on $\cal N$ one can extend the infinitesimal
variation of $\phi$ corresponding to a given section of $\phi^{*}T{\cal N}$ to
some finite variation. This construction determines a neighborhood of the
mapping $\phi$ being isomorphic to some open set (the restriction to some open
subset of the body) of the vector superspace of sections
$\Gamma(\phi^{*}T{\cal N})$.

G-structures on supermanifolds. Our next goal is to introduce the notion of
G-structure on a supermanifold. Let $\cal M$ be a supermanifold of dimension
$(m\oplus n|q)$ and let $\cal U$ be a covering of the body ${\cal M}_{R}$ such
that the restriction of $\cal M$ to any $U\in{\cal U}$ is isomorphic to a
superdomain $\rm U^{m\oplus n|q}$. For any $U,V\in{\cal U}$ denote by
$\phi_{V,U}$ the corresponding gluing transformation (from the chart $U$ to
$V$). We say that $L({\cal M})$ is (the total space of) the frame bundle over
$\cal M$ if $L({\cal M})$ is a supermanifold glued together from the
superdomains $U^{m\oplus n|q}\times GL(m\oplus n|q)$ by means of
transformations $\phi_{V,U}\times J$ where $J$ is the left multiplication by
the jacobian matrix corresponding to $\phi_{U,V}$. Explicitly, if
$X=(x^{1},\dots,x^{m},y^{1},\dots,y^{n},\theta^{1},\dots,\theta^{q})$ are
coordinates on the superdomain $\rm U^{m\oplus n|q}$ then the gluing
transformation from the chart $U^{m\oplus n|q}\times GL(m\oplus n|q)$ to the
chart $V^{m\oplus n|q}\times GL(m\oplus n|q)$ acts on the element $(a,g)\in\rm
U^{m\oplus n|q}_{\Lambda}\times GL_{\Lambda}(m\oplus n|q)$ as follows:
$(a,g)\mapsto(\phi_{V,U}(a),\frac{\partial\phi_{V,U}}{\partial X}(a)g)$. The
supermanifold $L({\cal M})$ constructed this way is equipped with a natural
structure of principal $GL(m\oplus n|q)$bundle over $\cal M$. One can show
that $L({\cal M})$ does not depend on the choice of covering $\cal U$. Let $G$
be a Lie (super)subgroup of $GL(m\oplus n|q)$ then, any smooth mapping ${\cal
M}\rightarrow G$ acts naturally from the right on $L({\cal M})$. Furthermore,
the superspace of maps $G^{\cal M}$ can be furnished with a natural supergroup
structure (in a “pointwise” manner). This supergroup acts on $L({\cal M})$.
Taking the quotient space with respect to this action one obtains a superspace
denoted by $L({\cal M})/G^{\cal M}$. Since the elements of $G^{\cal M}$ act
fiberwise , the superspace $L({\cal M})/G^{\cal M}$ is equipped with a
structure of fiber bundle over $\cal M$. For any Lie (super)subgroup $G\subset
GL(m\oplus n|q)$ we say that there is a $G$-structure defined on a
supermanifold $\cal M$ if a section of the bundle $L({\cal M})/G^{\cal
M}\rightarrow{\cal M}$ is specified. In general, given a bundle $p:{\cal
E}\rightarrow{\cal B}$ where ${\cal E},{\cal B}$ one can consider the
corresponding superspace of sections denoted by $\Gamma({\cal E})$ as a
subspace in $\cal E^{B}$. Thus we can consider the superspace of sections of
the bundle $L({\cal M})/G^{\cal M}\rightarrow{\cal M}$. We call this
superspace the superspace of $G$-structures over $\cal M$. The supergroup of
diffeomorphisms of $\cal M$ acts naturally on the superspace of $G$-structures
over $\cal M$. The corresponding quotient space is called the moduli space of
$G$-structures. Note that this moduli space is a superspace by definition but
in general it is not a supermanifold. In particular situations the most
interesting case is that of locally standard $G$-structures. For a
$G$-structure to be locally standard we mean the following . Fix a
$G$-structure $B_{st}$ on ${\bf R}^{m+n|q}$ that will be called standard. Note
that a $G$-structure on supermanifold $\cal M$ induces a $G$-structure on any
superspace obtained by restriction to a subset of the body $U\subset{\cal
M}_{R}$. Assume $\cal U$ is a standard atlas for the manifold $\cal M$, i.e.
$\cal M$ restricted to any element from $\cal U$ is equivalent to a
superdomain. Then, a $G$ structure $B$ on $\cal M$ is called locally standard
if the restriction of $B$ to any neighborhood $U\in{\cal U}$ is isomorphic to
the restriction of $B_{st}$ to $U$.

A rigorous definition of $(p,q)$-superconformal manifold can also be given in
the language of $G$-structures. This would provide a rigorous definition for
the moduli space of $(k\oplus l|q)$-dimensional supermanifolds as was outlined
above for the general case of arbitrary $G$-structure.

## References

  * [1] A. Konechny, A. Schwarz On $(k\oplus l|q)$-dimensional supermanifolds. , to appear in proceedings of international seminar ”Supersymmetry and quantum field theory” Kharkov (1997), Springer-Verlag.
  * [2] G. Moore, P. Nelson, J. Polchinsky Strings and supermoduli Phys. Lett. B 169 (1986), pp. 47-53
  * [3] A. Schwarz , On the definition of superspace Teor. Mat. Fiz. 60 (1984), p. 37; English transl. in Theoret. and Math. Phys. 60 (1984)
  * [4] A. Voronov, Mappings of supermanifolds Teor. Mat. Fiz. 60 (1984), p. 43
  * [5] A. Schwarz , Superanalogs of symplectic and contact geometry and their applications to quantum field theory , Amer. Math. Soc. Transl. (2), 177 (1996) , pp. 203-218
  * [6] G. Segal The definition of conformal field theory, preprint; Two dimensional conformal field theories and modular functors, in IXth International Conference on Mathematical Physics, B. Simon, A. Truman and I.M. Davies Eds. (Adam Hilger, Bristol, 1989).
  * [7] A. Konechny, A. Schwarz Geometry of N=1 super Yang-Mills theory in curved superspace Jr. of Geom. and Phys. (1997); hep-th/9609081
  * [8] P. Deligne Real versus complex , IAS QFT school notes (1997).
  * [9] A. Schwarz Topology for physicists Springer-Verlag Berlin Heidelberg 1994