We propose a candidate c -function in arbitrary dimensional quantum field theories with broken Lorentz and rotational symmetry. For holographic theories we derive the necessary and sufficient conditions on the geometric background for these c -functions to satisfy the c -theorem. We obtain the null energy conditions for anisotropic background to show that they do not themselves assure the c -theorem. By employing them, we find that it is possible to impose conditions on the UV data that are enough to guarantee at least one monotonic c -function along the RG flow. These UV conditions can be used as building blocks for the construction of anisotropic monotonic RG flows. Finally, we apply our results to several known anisotropic theories and identify the region in the parameter space of the metric where the c -theorem holds for our proposed c -function.
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