TODO 6VZMW Add topological derivation interface for anomaly cancellation conditions

[ValentinBredemestre]

Introduce a TopologicalDerivation structure recording anomaly cancellation conditions as linear, quadratic and cubic homogeneous forms arising from a formal derivation (e.g. triangle diagrams or index-theoretic methods).

Provide a canonical map from TopologicalDerivation to ACCSystem, together with basic simp lemmas, and update the document structure accordingly.

[jstoobysmith]

Is there a difference between ACCSystem and TopologicalDerivation here? If not, I'm not sure we need the latter.

[ValentinBredemestre]

Sorry for the late reply, I’ve been quite busy recently. : /

This is my first time handling a TODO in the project, so I’m still learning the conventions and best practices here. Thanks a lot for the comment and for pointing this out, I really appreciate it. = )

I’ve updated the code to avoid duplicating ACCSystem and instead introduced TopologicalDerivation as a lightweight abbreviation, whose purpose is only to record the intended origin of the anomaly cancellation conditions. I hope this aligns better with the overall design. = )

[jstoobysmith]

No worries :). This is probably one of the harder TODOs one could have picked :). I think what is probably best, because the documentation you have included is nice is the following:

1. Keep the TODO item in place, but add more detail based on your documentation for the "## J" subheading.
2. Don't define TopologicalDerivation explicitly, but add some information to the documentation string of ACCSystem stating that it is what should be derived (etc.).

If this makes sense.

[ValentinBredemestre]

Thanks a lot, that makes sense. = )

Before working on TODO 6VZMW more concretely, I just wanted to check what level of formalization you would expect there. In particular, would you envision something like a lightweight abstract structure encoding “gauge data” together with a map producing an ACCSystem, or is the intention more to keep this purely at the level of documentation for now?

I want to make sure any further work aligns well with the intended scope of this file and the project. = )

[jstoobysmith]

Awesome, thanks for your edits.

In particular, would you envision something like a lightweight abstract structure encoding “gauge data” together with a map producing an ACCSystem, or is the intention more to keep this purely at the level of documentation for now?

I think the former. Ideally one would have a data type corresponding to a gauge field, and then be able to construct the triangle diagrams or use the topological index theorems to get from that to the ACC conditions. There are likely easier intermediate steps that could be done here though and maybe a start would be to break this TODO down into smaller TODO items.

[jstoobysmith]

Approved - and will merge now :).

[jstoobysmith]

More generally, in this direction is this GitHub issue #880 which involves refactoring all of the code around these anomaly cancellation conditions, to make them centered on a key data structure.