Regarding the duality of the XY-placket model in the $2+1d$ system, We
compared and discussed the equivalence and difference between the modified
Villain formulation, recently introduced by Gorantla et al. to study duality in
exotic field theories such as fracton theory, and the standard Villain
formulation and dual Hamiltonian method we introduced in our previous paper to
study duality between Josephson junctions and quantum phase slips in the $1+1d$
system.
