Particle-vortex duality and theta terms in AdS/CMT applications

Abstract

In this paper we study particle-vortex duality and the effect of theta terms from the point of view of AdS/CMT constructions. We can construct the duality in 2+1 dimensional field theories with or without a Chern-Simons term, and derive an effect on conductivities, when the action is viewed as a response action. We can find its effect on 3+1 dimensional theories, with or without a theta term, coupled to gravity in asymptotically AdS space, and derive the resulting effect on conductivities defined in the spirit of AdS/CFT. AdS/CFT then relates the 2+1 dimensional and the 3+1 dimensional cases naturally. Quantum gravity corrections, as well as more general effective actions for the abelian vector, can be treated similarly. We can use the fluid/gravity correspondence, and the membrane paradigm, to define shear and bulk viscosities η and ζ for a gravity plus abelian vector plus scalar system near a black hole, and define the effect of the S-duality on it. * 1 See e.g., [21,22] for an early example of 2+1 dimensional duality in the path integral.

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

“…In any case, the α ij and κ ij coefficients are invariant under changes of the eletric/magnetic variables (S-duality). The other formulas are consistent with previous results at ρ = B = 0, where we know the effect of S-duality [9].…”

“…It was shown in [9] that this transformation comes from a simple duality transformation on the action (going to a master action and then writing a dual action in terms of a previously auxiliary field). Moreover, as we can see, since σ xx = Z(r H ) and σ xy = −W (r H ), this transformation becomes…”

“…The question we want to ask is, what is the effect of S-duality on this bulk holographic theory on the transport coefficients for the holographic dual field theory? The S-duality should correspond to particle-vortex duality in the boundary [8,9]. We will not consider the effect of quantum gravitational corrections to the bulk gravity action (those have been addressed in [9]).…”

“…The S-duality should correspond to particle-vortex duality in the boundary [8,9]. We will not consider the effect of quantum gravitational corrections to the bulk gravity action (those have been addressed in [9]). Since we are after the effect of S-duality, we will consider a vector action that involves both F µν and its dualF µν .…”

A renormalisation group equation for transport coefficients in (2 + 1)-dimensions derived from the AdS/CMT correspondence

“…A static magnetic field normal to the two space directions on the boundary can be incorporated by giving the black-hole a magnetic charge, promoting it to the status of a dyon in (3 + 1)-dimensions. This setup gives a promising framework in which to study the quantum Hall effect and there has been a number of papers examining this possibility [4]- [24].…”