  There are two main universality classes for depinning of elastic interfaces
in disordered media: quenched Edwards-Wilkinson (qEW), and quenched
Kardar-Parisi-Zhang (qKPZ). The first class is relevant as long as the elastic
force between two neighboring sites on the interface is purely harmonic, and
invariant under tilting. The second class applies when the elasticity is
non-linear, or the surface grows preferentially in its normal direction. It
encompasses fluid imbibition, the Tang-Leschorn cellular automaton of 1992
(TL92), depinning with anharmonic elasticity (aDep), and qKPZ. While the field
theory is well developed for qEW, there is no consistent theory for qKPZ. The
aim of this paper is to construct this field theory within the Functional
renormalization group (FRG) framework, based on large-scale numerical
simulations in dimensions $d=1$, $2$ and $3$, presented in a companion paper.
In order to measure the effective force correlator and coupling constants, the
driving force is derived from a confining potential with curvature $m^2$. We
show, that contrary to common belief this is allowed in the presence of a KPZ
term. The ensuing field theory becomes massive, and can no longer be Cole-Hopf
transformed. In exchange, it possesses an IR attractive stable fixed point at a
finite KPZ non-linearity $\lambda$. Since there is neither elasticity nor a KPZ
term in dimension $d=0$, qEW and qKPZ merge there. As a result, the two
universality classes are distinguished by terms linear in $d$. This allows us
to build a consistent field theory in dimension $d=1$, which loses some of its
predictive powers in higher dimensions.
