Yeah, no worries, it's late.

@Danu David Bar Moshe raises an interesting point on the Philbin paper, it's got the generalizations in eqs 21-22 arxiv.org/pdf/1303.0687v3.pdf

@EmilioPisanty Yeah! We hadn't seen that!

Different way to arrive at an almost identical result!

Also, I discovered that the equation 6.30 does not uniquely determine the variation of the potentials

One can take a different variation of the potentials (I'll give it in my answer) and hence get different conserved currents too---though they may be trivial/uninteresting?

The current for the other transformation is:

$$ G^{\mu\nu}\partial_{\alpha_1}\cdots \partial_{\alpha_n}A_\mu- (G\leftrightarrow F, A\leftrightarrow C)$$

@Danu Yeah, that's probably true. The question is whether your new variation isn't related to the old one by a U(1) gauge transform.

@EmilioPisanty Hmmmmmmmm... :P

Well, the transformation is

versus:

Looks pretty different to me!

...but maybe only superficially so!

Lemme think

Is there some reason to disregard the first?

The target is $$\delta A_\mu=-\xi^{\alpha_1\cdots \alpha_n}\partial_{\alpha_1}\cdots \partial_{\alpha_{n-1}} G_{\alpha_n \mu} \qquad \qquad \delta C_\mu=\xi^{\alpha_1\cdots \alpha_n}\partial_{\alpha_1}\cdots \partial_{\alpha_{n-1}} F_{\alpha_n \mu}$$

Focus on $$\delta C_\mu=\xi^{\alpha_1\cdots \alpha_n}\partial_{\alpha_1}\cdots \partial_{\alpha_{n-1}} F_{\alpha_n \mu}$$

Oh, right, gauge invariant!

The current is not gauge invariant

$$\delta C_\mu=\xi^{\alpha_1\cdots \alpha_n}\partial_{\alpha_1}\cdots \partial_{\alpha_{n-1}} (\partial_{\alpha_n}A_\mu-\partial_\mu A_{\alpha_n})$$

right?

...right

So $$\delta C_\mu^\text{you}-\delta C_\mu^\text{they}=\partial_\mu\left[\xi^{\alpha_1\cdots \alpha_n}\partial_{\alpha_1}\cdots \partial_{\alpha_{n-1}} A_{\alpha_n}\right]$$

yeah, okay

So your transformed potential is a gauge transformed version of theirs.

As it must because it gives the same force fields.

mhm

However, their current is gauge invariant

while mine will not be that

I think I basically ended up fixing a gauge

or maybe I even somehow left the surface of constraint inadvertently

@Danu Hang on, isn't it? Is it meant to?

These are the zilch and so on, right?

@EmilioPisanty Their current is gauge invariant right, because it consists of just $F$'s and $G$'s

yes

6.31 and 6.33

But the one coming from the transformation I propose is not gauge invariant

@Danu Huh, that's bizarre

What's your current look like?

@EmilioPisanty Or is it? :P

I gave it above

It has the A and the C appearing explicitly, of course

oh, ok

It could still potentially be gauge invariant?

Probably not, though

Yea.. I guess, but not manifestly

I think I should probably check if the constraint is even allowed (doesn't levae the constraint surface)...

@Danu Yeah, that's a good place to go. You might be breaking $G=*F$ at some point.

@Danu Yeah, that's definitely not gauge invariant.

@EmilioPisanty Also I think it obviously fails to be on the constraint surface

the trick of the trafo in the paper is that by the E.O.M. the variation doesn't contribute

Oh wait

sorry that's nonsense and confusing, disregard that

@Danu Does it? It reproduces the correct variation in the fields, right? 6.30?

@EmilioPisanty Yeah, yeah

I guess it's just a gauge-fixed version of their variation

Actually, I like this little learning process: I'll incorporate it in my answer.

@Danu I'm still a bit confused about exactly what's going on but I'm looking forward to that.

@EmilioPisanty I'm not going to be saying anything profound about it :P

Don't get your hopes up ;)

The answer is getting way, way long anyways...

@Danu Yeah, that was probably always going to happen

But there's a point at which it just becomes whooooa. Sort of.

I guess we should be happy we found an answer at all :)

@Danu Yeah, I'm really very happy with what we've found.

...getting there!