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In this paper we study the Wiedemann-Franz laws for transport in 2+1dimensions, and the action of $Sl(2,\mathbb{Z})$ on this transport, fortheories with an AdS/CMT dual. We find that $Sl(2,\mathbb{Z})$ restricts theRG-like flow of conductivities and that the Wiedemann-Franz law is $\bar L=\bar\kappa/(T\sigma)=cg_4^2\pi/3$, from the weakly coupled} gravity dual. In aself-dual theory this value is also the value of $L =\kappa/(T\sigma)$ in theweakly coupled field theory description. Using the formalism of a 0+1dimensional effective action for both generalized $SYK_q$ models and the$AdS_4$ gravity dual, we calculate the transport coefficients and show how theycan be matched at large $q$. We construct a generalization of this effectiveaction that is invariant under $Sl(2,\mathbb{Z})$ and can describe vortexconduction and integer quantum Hall effect.

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