  A comprehensive theory of the Kosterlitz-Thouless transition in
two-dimensional superfluids in thermal equilibrium can be developed within a
dual representation which maps vortices in the superfluid to charges in a
Coulomb gas. In this framework, the dissociation of vortex-antivortex pairs at
the critical temperature corresponds to the formation of a plasma of free
charges. The physics of vortex unbinding in driven-dissipative systems such as
fluids of light, on the other hand, is much less understood. Here we make a
crucial step to fill this gap by deriving a transformation that maps the
compact Kardar-Parisi-Zhang (KPZ) equation, which describes the dynamics of the
phase of a driven-dissipative condensate, to a dual electrodynamic theory. The
latter is formulated in terms of modified Maxwell equations for the
electromagnetic fields and a diffusion equation for the charges representing
vortices in the KPZ equation. This mapping utilizes an adaption of the Villain
approximation to a generalized Martin-Siggia-Rose functional integral
representation of the compact KPZ equation on a lattice.
