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Why dyon and monopole are mutually non-local objects?

This statement from Matteo Bertolini: Lectures on Supersymmetry, section 12.3.3 and related to this question.

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A good reference that works this out in some detail is Zwanziger's 1968 article, Quantum field theory of particles with both electric and magnetic charges. He introduces dual vector potentials for electric and magnetic fields, and shows that there are necessarily nonlocal terms in the Hamiltonian, and the canonical commutation relations between the fields and potentials. A typical example of such a commutation relation is the following, $$[\vec{F}_i^{\alpha}(x), \vec{V}_j^{\beta}(x')]=i\delta^{\alpha\beta}\delta_{ij}\delta(x-x')-i\delta^{\alpha\beta}\nabla_i\nabla_j\frac{1}{4\pi|x-x'|}$$ Here $F(x)$ refers to a vector constructed out of the electric and magnetic field strengths, and $V(x)$ a vector constructed out of the potentials. In general, two fields are mutually nonlocal if they do not commute at spacelike separations. For such fields, we cannot write a manifestly local Lagrangian that describes their interaction. In gauge theories with electric-magnetic duality, we can choose a convenient duality frame, write down a potential, and a corresponding local Lagrangian for one of the fields, and incorporate the other through boundary conditions, as in the case of how 't Hooft defect lines are described in the language of electric vector potentials.

The best one can do in this example is to introduce auxiliary parameters in such a way that the nonlocal terms vanish everywhere in space except on a Dirac string. $$[\mathcal{A}_i(x),\mathcal{B}_j(x')]=-i\epsilon_{ijk}h_k(x-x')$$ where $h(x)$ is the Dirac string function, and $\mathcal{A},\mathcal{B}$ are the electric and magnetic potentials. We can move the Dirac string around by redefining the potentials, a particular choice, when the string is along the z-axis, is $h_z(x)=-(1/2)\hat{z}\epsilon(z)\delta(x)\delta(y)$. We then have manifestly local commutators everywhere except along the z-axis.

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    $\begingroup$ Articles on expectation values of 't Hooft line operators in 4d $\mathcal{N}=2$ theories often deal with choices of a convenient duality frame, although the language has changed somewhat from the Dirac-Schwinger-Zwanziger papers. I'll put more links in the comments if I can find a good review from one of them. $\endgroup$
    – NewUser
    Mar 30, 2020 at 23:38
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    $\begingroup$ Dyons can be described as purely electrically charged fields by appropriate field redefinitions. One chooses a dual description E'=Ecos$\theta$+Bsin$\theta$, B'=-Esin$\theta$+Bcos$\theta$, which changes the charge assignments $(e,g)$ to $(e',0)$. $\endgroup$
    – NewUser
    Mar 30, 2020 at 23:42
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    $\begingroup$ I believe Seiberg-Witten's SU(2) theory has been studied at the dyon point by the original authors, and the prepotential can be written in that frame. $\endgroup$
    – NewUser
    Mar 30, 2020 at 23:44
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    $\begingroup$ arxiv.org/pdf/1503.00499.pdf This paper works out a dyon-dyon interaction Lagrangian, so it might be helpful. In general, to couple two mutually nonlocal fields, you have to incorporate one as a boundary condition or defect and the other as a fundamental field in the Lagrangian. A reference for this would be the work of Gaiotto, Moore, Neitzke on BPS bound states. $\endgroup$
    – NewUser
    Mar 30, 2020 at 23:55
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    $\begingroup$ Yes, you could choose a potential for the dyon by changing frames, but then your monopole would have to be described as a topological object, since the two are mutually nonlocal. $\endgroup$
    – NewUser
    Mar 30, 2020 at 23:58

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