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We consider $CP^{N-1}$ models in $d+1$ dimensions around Lifshitz fixedpoints with dynamical critical exponent $z$, in the large-N expansion. It isshown that these models are asymptotically free and dynamically generate a massfor the $CP^{N-1}$ fields for all $d=z$. We demonstrate that, for $z=d=2$, theinitially nondynamical gauge field acquires kinetic terms in a way similar tousual $CP^{N-1}$ models in 1+1 dimensions. Lorentz invariance emergesgenerically in the low-energy electrodynamics, with a nontrivial dielectricconstant given by the inverse mass gap and a magnetic permeability which has alogarithmic dependence on scale. At a special multicritical point, thelow-energy electrodynamics also has $z=2$, and an essentially singulardependence of the effective action on $B=\epsilon_{ij}\partial_iA_j$.

$CP^{N-1}$ Models at a Lifshitz Point