Identity missing in TInvar?

[Sukruti]

Hello,

Identity (3.6) from the attached article 'Normal forms for tensor polynomials. I. The Riemann tensor', is not getting implemented by FullSimplification[] or RiemannSimplify (using the package TInvar). I first tried to simplify the expression using FullSimplification[] in the notebook 'Weyl Identity' and then using RiemannSimplify in 'RiemSimp Weyl Identity'. Neither gave 0.

Is the identity missing from TInvar and Invar?

Best,
Sukruti

[Barry Wardell]

Hi Sukruti,

That particular identity is dimension dependent and applies only in 4D and below. Dimension-dependent identities are not yet implemented in TInvar. The original Invar did include identities that apply in dimension 4, but those are only for scalars and Eq. (3.6) has free indices so is not included.

It would be possible to include dimension-dependent identities, but as far as I recall there is some extra complexity to this that has not been addressed yet.

Regards,

Barry

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[Teake Nutma]

Hi,

For what it's worth, it is possible to construct all dimensional dependent identities (DDIs), including equation 3.6 of Fulling et. al., with xTras' ConstructDDIs. The tutorial for the Gauss-Bonnet term shows how to proof that certain terms are zero when DDIs are taken into account, and it just so happens to use eq. 3.6 of Fulling et. al. (although in terms of Riemann tensors, not Weyl tensors).

However, showing that certain combinations are zero is easier than canonicalizing expressions like TInvar does. And I haven't yet analyzed how ConstructDDIs could play well nice with TInvar functionality. So depending on your use case, your mileage may vary.

Best,

Teake

[Sukruti]

Hello,

Thanks Barry and Teake!

@Barry, you say "The original Invar did include identities that apply in dimension 4, but those are only for scalars". I understand this to mean that Invar allows the application of 4D DDIs only via RiemannSimplify and not via FullSimplification, is it correct? I tried using FullSimplification[] on scalar expressions which include derivatives of a scalar field phi in them. The DDI did not get applied on those expressions via FullSimplification[]. The attached notebook shows what I'm talking about.

@Teake, the links you shared are very useful, thanks! However I'm not sure if it's correct to say "it is possible to construct all dimensional dependent identities (DDIs), ... with xTras' ConstructDDIs". ConstructDDIs works only if there are at least 2d number of indices in the input expression while allowing two of the indices to remain free (shown in the attached notebook).  'd' is the number of dimensions here. If the DDI expressions are required to have all the indices contracted, then ConstructDDIs works only if there are at least 2d+2 number of indices in the input expression.

Best,

Sukruti

[Teake Nutma]

Hi Sukruti,

Yes you're right, ConstructDDIs needs at least 2(d+1) indices. But they can be freely chosen between the indices of the input expression and the free indices. You can for instance do the following in four dimensions:
```
ConstructDDIs[CD[-a]@CD[-b]@RiemannCD[-c, -d, -e, -f], {-a, -b, -c, -d}]
```

ConstructDDIs will give you all DDIs that come from over-antisymmetrization. I think there are no other kinds of DDIs, but I'm a little rusty on the topic so I'm happy to be proven wrong. 

Note that the DDIs returned by ConstructDDIs are not necessarily irreducible. Some are linear combinations of each other or identically zero when multi-term symmetries are taken into account. In case you're interested in how to systematically construct and eliminate DDIs, have a look at the notebook I've had uploaded along with https://arxiv.org/abs/1404.7452. While that paper is about a very specific subject, the methods in the notebook might be generic enough to suit your use case.

Best,

Teake

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[Sukruti]

Dear Teake,

Thanks for the explanations!

Going through the tutorial for the Gauss-Bonnet term which you shared earlier, I thought that the redundant DDIs can be eliminated using SolveTensors as shown there.

But now when I try to get independent DDIs which include the Levi-Civita tensor, I notice that SolveTensors is giving multiple DDIs for a given term, which are different from each other. The attached notebook shows the issue.

I tried tackling it using the command xAct`xTras`Private`DeleteDuplicateFactors@Flatten@TwoDerPreDDI1 which I found in the notebook attached to your article https://arxiv.org/abs/1404.7452. But it doesn't seem to help here.

Any suggestions how to solve this issue?

Best,

Sukruti

[Teake Nutma]

Hi Sukruti,

For the particular case in the Gauss-Bonnet xTras tutorial, SolveTensors can indeed be used to eliminate some dependent DDIs. But this is a coincidence (which unfortunately isn’t stressed the wording) and doesn’t work in the general case. The reason it works in the tutorial is because there all DDIs are symmetric in the free indices. In your notebook that’s not the case, and indeed you have DDIs where the free indices are just permutations of other DDIs. DeleteDuplicateFactors doesn’t help you there, since that only eliminates duplicates that differ by an overall constant.

ConstructDDIs is really for enumerating DDIs, not for generating an irreducible basis of DDIs. If you’re trying to achieve the latter, then I’m afraid there’s nothing in xTras or my own tool belt that helps towards that.

Best,

Teake

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<DDIs with Levi-Civita.nb>

[Sukruti]

Dear Teake,

Thank you for your reply. You said "For the particular case in the Gauss-Bonnet xTras tutorial, SolveTensors can indeed be used to eliminate some dependent DDIs. ... The reason it works in the tutorial is because there all DDIs are symmetric in the free indices."

Does this mean that SolveTensors works correctly only if either the input expression does not have free indices or if the free indices are symmetric? If so, is SolveTensors expected to give incorrect output for expressions with one free index (like in the attached notebook) and when there's a Levi-Civita symbol present in the expressions (like in my previous notebook)? Or does SolveTensors work correctly but incompletely in such cases, i.e. it eliminates some redundant DDIs but not all of them?

Best,

Sukruti