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I am currently studying Seiberg Witten theory's strong and weak coupling spectrum from the paper of Ferrari-Bilal The Strong Coupling Spectrum of Seiberg Witten Theory. I have a couple of doubts regarding statements like this

Recall that $(1, −1) \in \mathcal{S}_{S,+}$ and $(1, 1) \in \mathcal{S}_{S,-}$ are the two different descriptions of the same section $p$ corresponding to one and the same dyon.

Here $\mathcal{S}_{S,+}$ ($\mathcal{S}_{S,-}$) is the strong coupling spectrum in the upper (lower) half plane of the moduli space. I realize they mathematically are equivalent local descriptions of the same section of the $SL(2, \mathbb{Z})$-bundle over the moduli space, however in a physical situation they are very different to me. Will the dyon in the spectrum have magnetic charge $1$ or $-1$? I feel this two options are very different and physically distinguishable, which would amount to distinguish whether we are in the upper or lower half plane of the moduli space and ultimately this would mean the $\mathbb{Z}_2$ symmetry would be broken.

Ferrari-Bilal make this following statement, which however does not clear me up on what one would measure in reality

We have learned that, though there is a unique spectrum $\mathcal{S}_S$ valid through all the region $\mathcal{R}_S$, we must introduce two different sets of couples $(n_e, n_m)$ to represent it. We will denote these two sets by $\mathcal{S}_{S,+}$ and $\mathcal{S}_{S,−}$.

On the same token what Ferrari-Bilal call democracy transformations, which on the original Seiberg Witten paper is accounted in section 5.5 (in the quote the ordering of the couples of charges is inverted with respect to Ferrari-Bilal, i.e. it is $(n_m, n_e)$)

It seems that we are seeing massless particles of charges $(1, 0)$ or $(1, −1)$. However, there is in fact a complete democracy among dyons. The BPS-saturated dyons that exist semiclassically have charges $(1, n)$ (or $(−1, −n)$) for arbitrary integer $n$. The monodromy at infinity brings about a shift $(1, n) \rightarrow (1, n−2)$. If one carries out this shift $n$ times before proceeding to the singularity at $u = 1$ or $u = −1$, the massless particles producing those singularities would have charges $(1, −2n)$ and $(1, −1 − 2n)$, respectively.

This statement is again mathematically clear but in an hypothetical experiment, which of them will be massless, all of them?

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  • $\begingroup$ The "name" of things is a matter of conventions: you have two different types of charges, some of which you call electric and some of which you call magnetic. But what you call electric need not be the same as what I call electric, so the question "does this particle have magnetic charge +1 or -1?" is meaningless. One observer may claim that it is +1 and another that it is -1, and both would be correct. This is weird in our world because we do not observe magnetic charges, and therefore there is a canonical choice for what we call electric. But this is not true in a Seiberg-Witten world. $\endgroup$ Mar 9, 2023 at 21:18
  • $\begingroup$ I agree the sign of the charge is a matter of convention, however the relative sign matters. So since the transformations are not changing the relative sign you say, the sign itself is meaningless $\endgroup$
    – lucabtz
    Mar 10, 2023 at 11:13
  • $\begingroup$ However in the second bit of the question, regarding democracy transformations, also the absolute value of the electric charge of the dyons changes, I don't think here it is a matter of convention anymore $\endgroup$
    – lucabtz
    Mar 10, 2023 at 11:18
  • $\begingroup$ 'I agree the sign of the charge is a matter of convention' about this, slightly unrelated, in our world, where charge conjugation is not a symmetry because of weak interactions, the sign of charge is not actually a matter of convention, no? $\endgroup$
    – lucabtz
    Mar 10, 2023 at 11:22
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    $\begingroup$ Not quite a full answer, but an argument in favour of physical indistinguishability under these SL(2,Z)-transformation of charges I recall from somewhere. Consider the interaction of two BPS dyons (e1, g1) and (e2, g2) in the non-relativistic limit. The static electromagnetic force between them seems non-invariant under SL(2), but should be cancelled via interaction with scalar field, since the particles are BPS and their static energy is additive. On the other hand, the first dynamical correction (Lorenz force) is proportional to (e1g2-g1e2), which is an SL(2)-invariant combination. $\endgroup$ Mar 11, 2023 at 9:28

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