What are Mutually Non-local Particles?

Question

Hi,

In the context of Seiberg-Witten Theory, There are two types of particles:

• Mutually Local Particles: This means that at the vacuum manifold, one can use a coordinate (an Electric-Magnetic duality frame) on Coulomb branch such that all hypermultiplet have electric charges which couple to a theory written completely in terms of usual $U(1)$ photon coupled to electric sources;
• Mutually Non-Local Particles: This means that there is no such coordinate

I have several questions:

• What is non-local here? why this name?
• In the case of $SU(2)$ theory, at the two singularities it is said that a monopole and a dyon become massless,  Does this particles are in BPS spectrum of theory at the original duality frame (coordinate $a$) or it is in new frame ($a_D$ for monopole and $a-2a_D$ for dyons.)?
• Where the flatness of $Sp(2r;\mathbb{Z})$ comes from (for a $r$-dimensional complex Coulomb branch)?

Comments

''mutually (non)local'' should mean that the corresponding fields (don't) commute at mutually space-like arguments. For examples see, e.g., http://arxiv.org/abs/hep-th/9406118.

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@ArnoldNeumaier, I think I found the answer for the first part of question:

We know that Dirac-Schwinger-Zwanziger (DSZ) quantization condition is invariant under electric-magnetic duality group ($Sp(2,\mathbb{Z})$ in the $SU(2)$ case or $Sp(2r;\mathbb{Z})$ in general). This means that for DSZ:

                                                          $ g_1.q_2-g_2.q_1\in\mathbb{Z}$

There are two separate set of particles or excitation of fields:

• If DSZ condition is exactly $0$, then you can find an electric-magnetic duality frame in which all charges are electric;
• If DSZ condition is not zero, then you can't find such duality frame. This means that Lagrangian contains (if gauge symmetry is broken to $U(1)^r$) usual photons that coupled to electric sources and also dual photons coupled to magnetic sources, so in this sense, there is non-locality which means that you have a Lagrangian which contains fields which are non-local with respect to each other (e.g. usual photons see magnetic monopoles as a non-local source). I think it is very rare to have such Lagrangians. But in principle there are, because Coulomb branch is coordinatized by $(a,a_D)$ at different patches and for some patches you can have weakly coupled IR Lagrangians that are dependent on both of these coordinates.

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I think you could pose this as a self-answer to your question. Note that mutual nonlocality is typical for dual descriptions, and it is the existence of the latter that is rare.