Questions about forms in general dimension and missing wedge product

[Alejandro Jiménez]

Hello everyone,

I am working with xTerior in general dimensions, and I have found a couple of problematic situations:

1) [I think this is something that should be corrected] 

When apply a exterior derivative (covariant or not) to a product of a 0-form and a k-form (then, no wedge is needed), once the Leibniz rule is applied, no wedge appears between the derivative of the 0-form and the rest. See the attached file.

2) Differential forms with a symbolic (for example, dimension-dependent) degree.

If I need a (D-1)-form I can skip the problem defining its dual which is a 1 form, but some unavoidable errors arise when dealing with hodge stars [see attached file]. For example, in the case I show there the only thing that should be taken into account, basically, is the selection of the appropriate sign in order to revert the product. I know that solving this problem is much more complicated since ToCanonical cannot distinguish which rank is lower, 1 or D-4 for example (or whatever criteria it uses). The solution may be something like a way to tell ToCanonical "put the forms with weird rank at the end, with no special order" or a way to select a customised order... I do not know. 

I just wanted to report and share these problems. Does anyone have a solution for them?

Thanks a lot!

Alejandro Jiménez Cano

[Leo Stein]

Hi Alejandro,

Thanks for finding and reporting these bugs! I have created issues on the github page: #3, #4. Alfonso and I (or others!) will have to think about how to fix them. Here is a temporary solution.

The first bug (issue #3) stems from using Positive[] in some Diff[] code to check if factors are 0-forms or not. This is in Sec. 2.6 of xTerior.nb, in the cell below "We still need definition when acting on Times". Unfortunately, Positive[] doesn't know what to do with e.g. Positive[dimM-1], so those terms don't get detected as being non-0-degree. We need to be much more conservative about this assumption—it's safer to assume something is a non-0-degree form and fix that assumption later. Maybe we need to use =!=0 instead, but I haven't thought about the Mma evaluation very carefully.

One workaround is to teach Positive[] about your assumptions for the dimension of the manifold. You can use this definition:

Unprotect[Positive];
Positive[dimM] := True
Positive[dimM + k_Integer] := Simplify[dimM > -k, Assumptions -> {dimM > 1}]
Protect[Positive];

Notice that in the third line, I made the assumption that the dimension is greater than 1. That means d-1 forms will be understood as being non-0-degree, but not d-2 forms. If you want d-2 forms to have this assumption as well, then you change the assumption to dimM>2. Or, replace that definition with something else if you want all forms to be considered non-0-degree.

I have not yet thought about how to fix the canonicalization bug with forms of degree d-k, and I might not have the expertise to address it. It requires understanding the inner workings of the canonicalizer, which is the most complicated part of xTensor. The internal function AddedSign currently only expects values of +1, -1, or 0, but it's receiving a sign (-1)^dimM. This ought to be considered valid and treated as a constant, since ConstantQ[(-1)^dimM] evaluates to True. But, I haven't attempted to generalize this. Jose or any other ToCanonical expert, do you think this will work?

Best

Leo

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[Leo Stein]

PS Thanks to Alfonso for writing a bugfix for the first issue. xTerior is now at version {"0.9.1",{2019,5,17}} and contains this fix. The tarball on the xact.es web site is still behind, but you can either (i) clone directly from the xTerior repo, or (ii) grab the zip or tarball from this release.

[Alfonso García-Parrado Gómez-Lobo]

Some more additional comments about the second issue (issue #4) can be
found here:

https://github.com/xAct-contrib/xTerior/issues/4

Alfonso.

> > On Wed, May 15, 2019 at 11:32 AM Alejandro Jiménez <alejimcan@gmail

[Alejandro Jiménez]

Thank you so much for your effort!

Alejandro Jiménez Cano