The Internal Structure of the Deconstructed Dirac Monopole

Abstract

I study the internal structure of the Dirac magnetic monopole in a deconstructed $U(1)$ gauge theory. The deconstructed $U(1)$ gauge theory has a product $U(1)^N$ gauge group, which breaks down to the diagonal $U(1)_{\mathrm{diag}}$ gauge group. A linear superposition of the Dirac monopoles each from each $U(1)$ gauge group placed on top of each other in the 3D space constitutes a Dirac monopole with a unit magnetic charge under the unbroken $U(1)_{\mathrm{diag}}$ gauge group. However, the Dirac monopole in each $U(1)$ gauge group has a fractional magnetic charge of the unbroken $U(1)_{\mathrm{diag}}$ gauge group with the same sign, which makes these ``constituent'' Dirac monopoles repel each other. Therefore, for the ``composite'' Dirac monopole of the $U(1)_{\mathrm{diag}}$ gauge group to be stable, there must be attractive forces that counter the repulsive magnetic Coulomb forces. I argue that such attractive forces are provided by the Nielsen-Olesen type magnetic flux tubes of unbroken gauge groups. This internal structure of the composite Dirac monopole of $U(1)_{\mathrm{diag}}$ gauge group resembles the composite magnetic monopole found in the model constructed by Saraswat arXiv:1608.06951. I estimate the size of the Dirac monopole in $U(1)_{\mathrm{diag}}$ gauge group from the balance between the magnetic Coulomb forces and the forces from the tension of the magnetic flux tubes. Implications of the results for the Weak Gravity Conjecture are briefly discussed.