Let $A^{k}(U,{\mathbb{R}})$ be the space of smooth real differential forms on an open subset $U$ of $N_{\mathbb{R}}$, then a superform of bidegree $(p,q)$ on $U$ is an element of

$$A^{p,q}(U):=A^{p}(U,{\mathbb{R}})\otimes_{C^{\infty}(U)}A^{q}(U,{\mathbb{R}})=C^{\infty}(U)\otimes_{\mathbb{Z}}\Lambda^{p}M\otimes_{\mathbb{Z}}\Lambda^{q}M.$$

Formally, such a superform $\alpha$ may be written as

$$\alpha=\sum_{|I|=p,|J|=q}\alpha_{IJ}d^{\prime}x_{I}\wedge{d^{\prime\prime}x_{J}}$$

where $I$ (resp. $J$) consists of $i_{1}<\dots<i_{p}$ (resp. $j_{1}<\dots<j_{q}$), $\alpha_{IJ}\in C^{\infty}(U)$ and

$$d^{\prime}x_{I}\wedge{d^{\prime\prime}x_{J}}:=(dx_{i_{1}}\wedge\dots\wedge dx_{i_{p}})\otimes(dx_{j_{1}}\wedge\dots\wedge dx_{j_{q}}).$$

The wedge product is defined in the usual way on the space of superforms $A(U):=\oplus_{p,q\leq n}A^{p,q}(U)$. There is a canonical $C^{\infty}(U)$-linear isomorphism $J^{p,q}:A^{p,q}(U)\rightarrow A^{q,p}(U)$ obtained by switching factors in the tensor product. The inverse of $J^{p,q}$ is $J^{q,p}$. We call $\alpha\in A^{p,p}(U)$ symmetric if $J^{p,p}\alpha=\alpha$.

There is a differential operator $d^{\prime}:A^{p,q}(U)\rightarrow A^{p+1,q}(U)$ given by

$$d^{\prime}\alpha:=\sum_{|I|=p,|J|=q}\sum_{i=1}^{r}\frac{\partial\alpha_{IJ}}{\partial x_{i}}d^{\prime}x_{i}\wedge d^{\prime}x_{I}\wedge{d^{\prime\prime}x_{J}}.$$

This does not depend on the choice of coordinates as $d^{\prime}=d\otimes{\rm id}$ on $A^{p,q}(U)=A^{p}(U,{\mathbb{R}})\otimes_{\mathbb{Z}}\Lambda^{q}M$ is an intrinsic characterization using the classical differential $d$ on the space $A^{p}(U,{\mathbb{R}})$ of real smooth $p$-forms. Similarly, we define a differential operator $d^{\prime\prime}:A^{p,q}(U)\rightarrow A^{p,q+1}(U)$ by

$$d^{\prime\prime}\alpha:=\sum_{|I|=p,|J|=q}\sum_{j=1}^{r}\frac{\partial\alpha_{IJ}}{\partial x_{j}}{d^{\prime\prime}x_{j}}\wedge d^{\prime}x_{I}\wedge{d^{\prime\prime}x_{J}}.$$

By linearity, we extend these differential operators to $A(U)$. Moreover, we set $d:=d^{\prime}+d^{\prime\prime}$.

If $N^{\prime}$ is a free abelian group of rank $r^{\prime}$ and if $F:N^{\prime}_{\mathbb{R}}\rightarrow N_{\mathbb{R}}$ is an affine map with $F(V)\subset U$ for an open subset $V$ of $N^{\prime}_{\mathbb{R}}$, then we have a well-defined pull-back $F^{*}:A^{p,q}(U)\rightarrow A^{p,q}(V)$ given as usual. The affine pull-back commutes with the differential operators $d$, $d^{\prime}$ and $d^{\prime\prime}$.

Let $A_{c}(U)$ denote the space of superforms on $U$ with compact support in $U$. For $\alpha\in A_{c}(U)$, we define

$$\int_{U}\alpha:=\int_{U}\alpha_{LL}dx_{1}\wedge\dots\wedge dx_{r}$$

with $L=\{1,\dots,r\}$ and the usual integration of $r$-forms with respect to the orientation induced by the choice of coordinates on the right hand side. If $F$ is an affine map as in 2.3 and if $r=r^{\prime}$, then we have the transformation formula

$$\int_{V}F^{*}(\alpha)=|\det(F)|\int_{U}\alpha$$

(1)

(see [La12], equation (2.3)). We conclude that the definition of the integral depends only on the underlying integral ${\mathbb{R}}$-affine structure of $N_{\mathbb{R}}$.

Now let $\sigma$ be a polyhedron of dimension $n$ in $N_{\mathbb{R}}$. By definition, $\sigma$ is the intersection of finitely many halfspaces $H_{i}:=\{\omega\in N_{\mathbb{R}}\mid\langle u_{i},\omega\rangle\leq c_{i}\}$ with $u_{i}\in M_{\mathbb{R}}$ and $c_{i}\in{\mathbb{R}}$. A polytope is a bounded polyhedron. We say that $\sigma$ is an integral $G$-affine polyhedron for a subgroup $G$ of ${\mathbb{R}}$ if we may choose all $u_{i}\in M$ and all $c_{i}\in G$. In this case, we have a canonical integral ${\mathbb{R}}$-affine structure on the affine space ${\mathbb{A}}_{\sigma}$ generated by $\sigma$. If ${\mathbb{L}}_{\sigma}$ is the underlying real vector space of ${\mathbb{A}}_{\sigma}$, then this integral structure is given by the lattice $N_{\sigma}:={\mathbb{L}}_{\sigma}\cap N$. Using 2.3 and the above, we get a well-defined integral $\int_{\sigma}\alpha$ for any $\alpha\in A_{c}^{n,n}(U)$, where $U$ is an open neighbourhood of $\sigma$.

In [CD12], integration is described in terms of a contraction: Similarly as in differential geometry, we may view a superform $\alpha\in A^{p,q}(U)$ as a multilinear map

$$N_{\mathbb{R}}^{p+q}\longrightarrow C^{\infty}(U),\quad(n_{1},\dots,n_{p+q})\mapsto\alpha(n_{1},\dots,n_{p+q})$$

which is alternating in the variables $(n_{1},\dots,n_{p})$ and also in $(n_{p+1},\dots,n_{p+q})$. Let $I\subset\{1,\dots,p+q\}$ be a subset of cardinality $s$ with $s^{\prime}$ elements contained in $\{1,\dots,p\}$ and hence $s^{\prime\prime}=s-s^{\prime}$ elements in $\{p+1,\dots,p+q\}$. Given vectors $v_{1},\dots,v_{s}\in N_{\mathbb{R}}$, the contraction $\langle\alpha;v_{1},\dots,v_{s}\rangle_{I}\in A^{p-s^{\prime},q-s^{\prime\prime}}(U)$ is given by inserting $v_{1},\dots,v_{s}$ for the variables $(n_{i})_{i\in I}$ of the above multilinear function.

Using the basis $e_{1},\dots,e_{r}$ of $N$ and assuming $\alpha\in A_{c}^{r,r}(U)$, the contraction $\langle\alpha;e_{1},\dots,e_{r}\rangle_{\{r+1,\dots,2r\}}$ is a $(r,0)$-superform which may be viewed as a classical $r$-form on $U$. Then it is immediately clear from the definitions that we have

$$\int_{U}\alpha=\int_{U}\langle\alpha;e_{1},\dots,e_{r}\rangle_{\{r+1,\dots,2r\}}$$

where we use the usual integration of $r$-forms on the right. Of course, there is no preference to contract with respect to the last $r$ variables. Similarly, may view $\langle\alpha;e_{1},\dots,e_{r}\rangle_{\{1,\dots,r\}}\in A_{c}^{0,r}(U)$ as a classical $r$-form and we have

$$\int_{U}\alpha=\int_{U}\langle\alpha;e_{1},\dots,e_{r}\rangle_{\{1,\dots,r\}}.$$

Let $H$ be an integral ${\mathbb{R}}$-affine halfspace in $N_{\mathbb{R}}$. This means that $H=\{\omega\in N_{\mathbb{R}}\mid\langle u,\omega\rangle\leq c\}$ for some $u\in M$ and $c\in{\mathbb{R}}$. Using a translation, we may assume that $c=0$ and hence the boundary $\partial H$ is a linear subspace of $N_{\mathbb{R}}$. Let $[\omega_{\partial H,H}]$ be the generator of $N/(N\cap\partial H)\cong{\mathbb{Z}}$ which points outwards, i.e. there is $u_{\partial H,H}\in M$ such that $u_{\partial H,H}(H)\leq 0$ and $u_{\partial H,H}(\omega_{\partial H,H})=1$. We choose a representative $\omega_{\partial H,H}\in N$ and we note also that $u_{\partial H,H}$ is uniquely determined by the above properties.

Let $U$ be an open subset of $N_{\mathbb{R}}$ and let $\sigma$ be an $r$-dimensional integral ${\mathbb{R}}$-affine polyhedron contained in $U$. For any closed face $\rho$ of codimension $1$, let $\omega_{\rho,\sigma}:=\omega_{\partial H,H}$ using 2.7 for the affine hyperplane $\partial H$ generated by $\rho$ and the corresponding halfspace containing $\sigma$. We note that $\omega_{\rho,\sigma}\in N$ is determined up to addition with elements in $N_{\rho}=N\cap{\mathbb{L}}_{\rho}$, where ${\mathbb{L}}_{\rho}$ is the linear hyperplane parallel to $\rho$.

For $\eta\in A_{c}^{r-1,r}(U)$, we have introduced the contraction $\langle\eta;\omega_{\rho,\sigma}\rangle_{\{r\}}$ as an element of $A_{c}^{r-1,r-1}(U)$ which is obtained by inserting the vector $\omega_{\rho,\sigma}$ for the $r$-th argument of the corresponding multilinear function (see 2.6). Note that the restriction of this contraction to $\rho$ does not depend on the choice of the representative $\omega_{\rho,\sigma}$. Then we define

$$\int_{\partial\sigma}\eta:=\sum_{\rho}\int_{\rho}\langle\eta;\omega_{\rho,\sigma}\rangle_{\{r\}},$$

where $\rho$ ranges over all closed faces of $\sigma$ of codimension $1$. On the right, we use the integrals of $(r-1,r-1)$-superforms from 2.4. For $\eta\in A_{c}^{r,r-1}(U)$, we define similarly

$$\int_{\partial\sigma}\eta:=\sum_{\rho}\int_{\rho}\langle\eta;\omega_{\rho,\sigma}\rangle_{\{1\}}.$$

Note that the integrals do depend only on the integral ${\mathbb{R}}$-affine structure of $N_{\mathbb{R}}$ but do not depend on the choice of the orientation of $N_{\mathbb{R}}$.

If $\sigma$ is an integral ${\mathbb{R}}$-affine polyhedron of any dimension $n$ and if $\eta\in A_{c}^{n-1,n}(U)$ for an open subset $U$ of $N_{\mathbb{R}}$ containing $\sigma$, then we define $\int_{\partial\sigma}\eta$ by applying the above to the affine space ${\mathbb{A}}_{\sigma}$ generated by $\sigma$ and to the pull-back of $\eta$ to ${\mathbb{A}}_{\sigma}$. We give now a concrete description of $\int_{\partial\sigma}\eta$ in terms of integrals over classical $n-1$-forms. For every closed face $\rho$ of $\sigma$, let $N_{\sigma}={\mathbb{L}}_{\sigma}\cap N$ be the canonical integral structure on the affine space generated by $\sigma$. If $e_{1}^{\rho},\dots,e_{n-1}^{\rho}$ is a basis of $N_{\rho}$, then $\omega_{\rho,\sigma},e_{1}^{\rho},\dots,e_{n-1}^{\rho}$ is a basis of $N_{\sigma}$. We note that the contraction $\langle\eta;\omega_{\rho,\sigma},e_{1}^{\rho},\dots,e_{n-1}^{\rho}\rangle_{\{n,\dots,2n-1\}}$ may be viewed as a classical $(n-1)$-form on $U$ and hence we get

$$\int_{\partial\sigma}\eta=\sum_{\rho}\int_{\rho}\langle\eta;\omega_{\rho,\sigma}\rangle_{\{n\}}=\sum_{\rho}\int_{\rho}\langle\eta;\omega_{\rho,\sigma},e_{1}^{\rho},\dots,e_{n-1}^{\rho}\rangle_{\{n,\dots,2n-1\}}.$$

A supercurrent on $U$ is a continuous linear functional on $A_{c}^{p,q}(U)$ where the latter is a locally convex vector space in a similar way as in the classical case. We denote the space of such supercurrents by $D_{p,q}(U)$. As usual, we define the linear differential operators $d$, $d^{\prime}$ and $d^{\prime\prime}$ on $D(U):=\bigoplus_{p,q}D_{p,q}(U)$ by using $(-1)^{p+q+1}$ times the dual of the corresponding differential operator on $A_{c}^{p,q}(U)$. The sign is chosen in such a way that the canonical embedding $A^{p,q}(U)\rightarrow D_{r-p,r-q}(U)$ is compatible with the operators $d$, $d^{\prime}$ and $d^{\prime\prime}$. Here, $\alpha\in A^{p,q}(U)$ is mapped to $[\alpha]\in D_{r-p,r-q}(U)$ given by $[\alpha](\beta)=\int_{N_{\mathbb{R}}}\alpha\wedge\beta$ for any $\beta\in A_{c}^{r-p,r-q}(U)$.