I'm trying to understand twisted differential forms. I do know that they are like regular differential forms but under coordinate transformations they pick up an extra factor of the sign of the determinant of the transformation. Somehow this means that they can be used to integrate on non-orientable manifolds. (???)
While Googling, I saw this question on Physics Forums, where one of the answers seems like a good start at understanding twisted differential forms. I'll quote the answer here for reference.
Consider a line segment. There are two ways one can orient this: along the segment and across the segment. For example, if you wanted to represent a segment of the world-line of a particle, then the first type of orientation is appropriate. On the other hand, imagine a circle drawn on a plane. A segment of this circle naturally has an orientation of the second type: it is oriented 'across' the segment, depending on which side of the circle is 'inside' and which is 'outside'
This is the main difference between differential forms and their twisted counterparts, i.e. the type of orientation.
The contour lines of a function have the 'across' orientation, and are represented by 1-forms. But if we wanted to represent coutour lines with an orientation along them instead of across, you would use a twisted 1-form.
Imagine 2+1 dimensional spacetime. I assume you're familiar with the usual picture of a 2-form in a three dimensional space. The 'tubes' or 'boxes' in the picture of this 2-form will have an orientation that is 'around' them, i.e. clockwise or anticlockwise. Of course, one can always convert from clockwise/anticlockwise to up/down using things like right-hand rules, but that is not the natural type of orientation of a current. For a twisted 2-form, on the other hand, the tubes or boxes will have the correct 'along' orientation. So, in 2+1 dimensional spacetime, current density is a twisted 2-form. Similarly, in 3+1 dimensions, it is a twisted 3-form.
Though this isn't the way I usually think of differential forms, I am somewhat familiar with the geometric interpretation — at least for $1$-forms — as stacks through which vectors penetrate. However I'm still not entirely able to see what a twisted differential $1$-form would be. Geometrically, is it supposed to be like a curve which "counts" the projections of the tangent vectors onto the tangents of the curve along it? If so, how is that picture obtained from the definition? And how does one visualize higher dimensional twisted forms — because I don't really understand that part of the post at all.
Best Answer
Integral of a twisted differential $p$-form on a $p$-submanifold $\Sigma \subset M$ does not depend on the orientiation of $\Sigma$ (in fact $\Sigma$ can be non-orientable), but rather on external orientation (orientation of the normal bundle $\left. TM \right|_\Sigma / T \Sigma$). To give you a physical example of this, consider the current $j$ of certain substance (e.g. water or electric charge) in a container $M$ (manifold of dimension $D$). I claim that $j$ is geometrically a twisted $(D-1)$-form on $M$. Indeed, given any two-sided hypersurface $\Sigma \subset M$, the flux $\int_\Sigma j$ of the flowing substance through $\Sigma$ is well-defined. Clearly it should not depend on the orientation of $\Sigma$, but it should change sign if we switch which side of $\Sigma$ is regarded as "in" vs "out".
Of course twisted $D$-forms are even easier: they represent densities and can be integrated over volume, without having to choose orientation. One example is the volume form: certainly given a metric you can calculate the area of a Mobius strip! To extend my previous physical analogy, suppose that the substance under consideration has density profile $\rho$. Then $\rho$ is a twisted $D$-form. If the amount of substance is (pointwise) conserved, we will have a continuity equation $$ \frac{\partial}{\partial t} \rho + d j=0, $$ in which $d$ is the exterior derivative along the "spatial" manifold $M$ and $t$ is additional "time" parameter. This equation is the statement that the rate of change of amount of the substance in some volume $\Omega$ is given by the flux of the substance through boundary $\partial \Omega$.