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Non-Abelian Josephson junction is a junction of non-Abelian colorsuperconductors sandwiching an insulator, or non-Abelian domain wall ifflexible, whose low-energy dynamics is described by a $U(N)$ principal chiralmodel with the conventional pion mass. A non-Abelian Josephson vortex is anon-Abelian vortex (color magnetic flux tube) residing inside the junction,that is described as a non-Abelian sine-Gordon soliton. In this paper, wepropose Josephson instantons and Josephson monopoles, that is, Yang-Millsinstantons and monopoles inside a non-Abelian Josephson junction, respectively,and show that they are described as $SU(N)$ Skyrmions and $U(1)^{N-1}$ vorticesin the $U(N)$ principal chiral model without and with a twisted mass term,respectively. Instantons with a twisted boundary condition are reduced (orT-dual) to monopoles, implying that ${\mathbb C}P^{N-1}$ lumps are T-dual to${\mathbb C}P^{N-1}$ kinks inside a vortex. Here we find $SU(N)$ Skyrmions areT-dual to $U(1)^{N-1}$ vortices inside a wall. Our configurations suggest a yetanother duality between ${\mathbb C}P^{N-1}$ lumps and $SU(N)$ Skyrmions aswell as that between ${\mathbb C}P^{N-1}$ kinks and $U(1)^{N-1}$ vortices,viewed from different host solitons. They also suggest a duality betweenfractional instantons and bions in the ${\mathbb C}P^{N-1}$ model and those inthe $SU(N)$ principal chiral model.

Josephson instantons and Josephson monopoles in a non-Abelian Josephson junction