integration over supermanifolds, Berezin integral, fermionic path integral
Kontsevich integral, Selberg integral, elliptic Selberg integral
Here we discuss the integration of a differential form (possibly twisted in some way) on a topological manifold (possibly with additional structure) over an appropriately structured submanifold (or formal linear combination thereof).
See differential form for basic definitions.
Let $X$ be an $n$-dimensional topological manifold, and let $\omega$ be a continuous $n$-pseudoform on $X$. Suppose that $X$ is paracompact and Hausdorff, so that we may find a locally finite cover of $X$ with a subordinate partition of unity and a continuous coordinate chart on each patch. (When $X$ is differentiable, or even smooth, then these may also be chosen to be differentiable or smooth, which may be convenient but is not necessary.) Then $\omega$ defines a measure on $X$ as follows:
On each coordinate patch $U$, fix the orientation given by the coordinates to turn $\omega$ into an untwisted $n$-form $\hat{\omega}$; then write $\hat{\omega}$ in coordinates as
In this situation, it is convenient also to write
in other words, we interpret $\mathrm{d}x^1 \cdots \mathrm{d}x^n$ as the absolute value of $\mathrm{d}x^1 \wedge \cdots \wedge \mathrm{d}x^n$.
The coordinates on $U$ define a diffeomorphism between $U$ and an open subset of $\mathbf{R}^n$ that we'll also call $U$; so use the latter formula to interpret
where the right-hand side is now interpreted in the usual way as as integral with respect to Lebesgue measure.
Using the partition of unity, write
where $w_U$ is a weight function defined on $U$ and $\omega_U$ is the restriction of $\omega$ to $U$. Then we have
or more generally,
for $E$ a measurable subset of $X$ and $\chi_E$ the characteristic function of $E$.
A priori, this definition depends not only on the particular coordinate patches chosen but also on the partition of unity chosen to go with them. Furthermore, the defintion could be done just as easily (perhaps even more easily) for something other than an $n$-pseudoform. But the (perhaps surprising) fact that justifies it all is this:
When $\omega$ is an $n$-pseudoform, the definition of $\int_E \omega$ is independent of the coordinates and partition chosen. Furthermore, the map from $n$-pseudoforms to measures is linear.
Note that, if $\omega$ were an $n$-form instead of a pseudoform, then the definition would depend on the orientation of the coordinates chosen. We could fix that by using the absolute value ${|\omega_U|}$ in place of $\omega_U$ in (1) and the following equations, but then the map from forms to measures would not be linear.
It may also be enlightening to consider how to go back from a measure to an $n$-pseudoform. If $\omega$ is an absolutely continuous Radon measure on $X$, then it defines an $n$-pseudoform (which we may also call $\omega$) as follows:
Again, this definition is independent of the coordinate system chosen (as long as it extends the given vectors); or if that's not true, then we messed up and need to add further restrictions to the absolutely continuous Radon measure $\omega$. The definition is not independent of the orientation chosen, of course; thus we get a pseudoform rather than an untwisted form. You might try to ignore the orientation and take $\omega(v_1,\ldots,v_n)$ to be $L$ always, but that does not define an exterior form, as is most easily seen if two vectors are switched (which does not change $L$). Instead, this would define an absolute differential form (which is equivalent to a pseudoform when, as here, the degree equals the dimension).
One can integrate forms other than $n$-pseudoforms, of course, but only over certain structures within the manifold $X$. Specifically, if $R$ is a $p$-dimensional submanifold of $X$ (that is a $p$-dimensional manifold $U$ equipped with a map $R\colon U \to X$), then we would like to integrate $p$-forms or $p$-pseudoforms (defined on $X$) over $R$. Here is how we do this:
We may integrate a $p$-form $\eta$ over $R$ if $R$ is oriented, that is if $U$ is oriented. We pull back $\eta$ from $X$ to $U$, then use the orientation on $U$ to turn $\eta$ into a $p$-pseudoform, which we can then integrate on the $p$-dimensional manifold $U$.
We may integrate a $p$-pseudoform $\eta$ over $R$ if $R$ is pseudooriented, that is if it is equipped with a map that, for each point $a$ on $U$, takes a local orientation of $X$ at $R(a)$ to a local orientation of $U$ at $a$, continuously in $a$ and taking opposite orientations to opposite orientations. Then locally, we turn $\eta$ into a $p$-form on $X$ using a local orientation on $X$, pull that back to $U$, and use the corresponding local orientation on $U$ to turn that back into a $p$-pseudoform, which we can then integrate on $U$.
Thus, while integration of $n$-pseudoforms is the most basic, integration of general $p$-forms is actually a bit simpler than integration of general $p$-pseudoforms. Integration of other twisted or vector-valued forms can also be done, again given appropriate structure on $R$.
Note that, if $X$ is thought of a submanifold of itself, then it has a natural pseudoorientation that takes each local orientation to itself, and so we recover the original definition of integration of $n$-pseudoforms on $X$. Also, if $X$ is an oriented manifold, then it may be viewed as an oriented submanifold of itself, giving a definition of integration of $n$-forms on $X$.
To integrate on unoriented submanifolds of arbitrary dimension, use absolute differential forms.
One often sees the definition of integration given for parametrised submanifolds, that is submanifolds where $U$ is an open subset of $\mathbf{R}^p$. This amounts to a combination of the concepts above, with the two uses of $U$ (as a coordinate patch in $X$ or as the source of a submanifold of $X$) identified. The theorem that the integral of an $n$-pseudoform on $X$ is independent of the coordinates chosen now becomes a theorem that the integral of a parametrised submanifold is independent of the parametrisation (up to some details about orientation), which in the end returns the result that one can integrate forms over arbitrary submanifolds (given an orientation or pseudoorientation as above).
We can also integrate on a formal linear combination of submanifolds, which is handy since these are the chains of de Rham homology?. This is straightforward:
An exposition with an eye towards applications in physics is section 3 of