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We present a detailed derivation of the renormalization group equations fortwo dimensional electromagnetic Coulomb gases whose charges lie on a triangularlattice (magnetic charges) and its dual (electric charges). The interactionsbetween the charges involve both angular couplings and a new electromagneticpotential. This motivates the denomination of ``elastic'' Coulomb gas. Suchelastic Coulomb gases arise naturally in the study of the continuous meltingtransition of two dimensional solids coupled to a substrate, eithercommensurate or with quenched disorder.

Electromagnetic Coulomb Gas with Vector Charges and "Elastic'' Potentials : Renormalization Group Equations