Wiedemann-Franz laws and Sl (2, ℤ) duality in AdS/CMT holographic duals and one-dimensional effective actions for them

In this paper we study the Wiedemann-Franz laws for transport in 2+1 dimensions, and the action of Sl (2 , ℤ) on this transport, for theories with an AdS/CMT dual. We find that Sl (2 , ℤ) restricts the RG-like flow of conductivities and that the Wiedemann-Franz law is L=κ/( Tσ)=cg_4^2 /3 , from the weakly coupled gravity dual. In a self-dual theory this value is also the value of L = κ /( Tσ ) in the weakly coupled field theory description. Using the formalism of a 0+1 dimensional effective action for both generalized SY K q models and the AdS 4 gravity dual, we calculate the transport coefficients and show how they can be matched at large q . We construct a generalization of this effective action that is invariant under Sl (2 , ℤ) and can describe vortex conduction and integer quantum Hall effect.