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

TL;DR

This paper explores the effects of $Sl(2, extbf{Z})$ duality on transport laws like Wiedemann-Franz in 2+1D theories with holographic duals, and develops an effective action framework for these phenomena, including quantum Hall effects.

Contribution

It introduces an $Sl(2, extbf{Z})$-invariant effective action formalism for holographic transport and extends the analysis of Wiedemann-Franz laws within this duality context.

Findings

$Sl(2, extbf{Z})$ constrains RG flow of conductivities

Wiedemann-Franz law value derived from gravity dual

Effective action matches transport coefficients at large $q$

Abstract

In this paper we study the Wiedemann-Franz laws for transport in 2+1 dimensions, and the action of $Sl(2,\mathbb{Z})$ on this transport, for theories with an AdS/CMT dual. We find that $Sl(2,\mathbb{Z})$ restricts the RG-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 a self-dual theory this value is also the value of $L =\kappa/(T\sigma)$ in the weakly coupled field theory description. Using the formalism of a 0+1 dimensional effective action for both generalized $SYK_q$ 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 $Sl(2,\mathbb{Z})$ and can describe vortex conduction and integer quantum Hall effect.