  We define a stochastic lattice model for a fluctuating directed polymer in
$d\geq 2$ dimensions. This model can be alternatively interpreted as a
fluctuating random path in 2 dimensions, or a one-dimensional asymmetric simple
exclusion process with $d-1$ conserved species of particles. The deterministic
large dynamics of the directed polymer are shown to be given by a system of
coupled Kardar-Parisi-Zhang (KPZ) equations and diffusion equations. Using
non-linear fluctuating hydrodynamics and mode coupling theory we argue that
stationary fluctuations in any dimension $d$ can only be of KPZ type or
diffusive. The modes are pure in the sense that there are only subleading
couplings to other modes, thus excluding the occurrence of modified
KPZ-fluctuations or L\'evy-type fluctuations which are common for more than one
conservation law. The mode-coupling matrices are shown to satisfy the so-called
trilinear condition.
