S-duality, entropy function and transport in AdS4/CMT3

Abstract

In this paper we consider Abelian vector plus scalar holographic gravity models for 2+1 dimensional condensed matter transport, and the effect of S-duality on them. We find the transport coefficients from the electric and heat currents via usual membrane paradigm-type calculations, and the effect of S-duality on them. We study the same system also by using the entropy function formalism in the extremal case, and the formalism of holographic Stokes equations, in the case of one-dimensional lattices. We study a few generalizations that appear when considering a supergravity-inspired model, and apply the entropy function method for them. *

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

“…We will also consider a slightly more general holographic setup describing a disordered system in the presence of a nontrivial electric charge density ρ and magnetic field B, as well as a nontrivial θF µνF µν term, corresponding to Chern-Simons in 3 dimensions, as in [9] (following [10] and [11]). We will analyze the Lorenz number L coming from that calculation, as well as the analogous law for the dissipative components xx, and the action of the Sl(2, Z) on the transport coefficients and the laws.…”

“…for the thermoelectric conductivities, where we restored the dependence on the Newton's constant κ 4 and of g 2 4 with respect to [9]. Here e 2V , Φ, Z and W are defined at the horizon r H of the black hole, with e 2V being the warp factor for the spatial part of the holographic boundary, (x, y).…”

“…+ e 2V (dx 2 + dy 2 )). But on the other hand, the entropy density of the field theory in (2κ 2 4 units), identified with the entropy of the black hole (Hawking formula) divided by the area in (x, y), gives also (see also [9])…”

“…On the other hand, the action of electromagnetic S-duality (understood as duality in 3+1 dimensions) on the same transport, is found from the equations derived in [9] for the holographic coefficients in the presence of ρ, B and W, Z in the dual, or in [20] in a similar calculation for holographic lattices. It corresponds in 2+1 dimensions to the same particle-vortex duality, and is denoted by an operation S.…”

“…It was noted in [9] that taking the limit ρ = B = 0 in formulas (3.2), while keeping Φ = 0, results in σ xx = Z/g 2 4 and σ xx = −4W . So indeed, S-duality corresponds via holography to particle-vortex duality in 3 dimensions, acting as we saw in (3.20).…”