  We show that Horava-Lifshitz gravity theory can be employed as a covariant
framework to build an effective field theory for the fractional quantum Hall
effect that respects all the spacetime symmetries such as non-relativistic
diffeomorphism invariance and anisotropic Weyl invariance as well as the gauge
symmetry. The key to this formalism is a set of correspondence relations that
maps all the field degrees of freedom in the Horava-Lifshitz gravity theory to
external background (source) fields among others in the effective action of the
quantum Hall effect, according to their symmetry transformation properties. We
originally derive the map as a holographic dictionary, but its form is
independent of the existence of holographic duality. This paves the way for the
application of Horava-Lifshitz holography on fractional quantum Hall effect.
Using the simplest holographic Chern-Simons model, we compute the low energy
effective action at leading orders and show that it captures universal
electromagnetic and geometric properties of quantum Hall states, including the
Wen-Zee shift, Hall viscosity, angular momentum density and their relations. We
identify the shift function in Horava-Lifshitz gravity theory as minus of
guiding center velocity and conjugate to guiding center momentum. This enables
us to distinguish guiding center angular momentum density from the internal
one, which is the sum of Landau orbit spin and intrinsic (topological) spin of
the composite particles. Our effective action shows that Hall viscosity is
minus half of the internal angular momentum density and proportional to Wen-Zee
shift, and Hall bulk viscosity is half of the guiding center angular momentum
density.
