Quantum critical transport, duality, and M theory

We consider charge transport properties of $2+1$ dimensional conformal field theories at nonzero temperature. For theories with only Abelian U(1) charges, we describe the action of particle-vortex duality on the hydrodynamic-to-collisionless crossover function: this leads to powerful functional constraints for self-dual theories. For $\mathcal{N}=8$ supersymmetric, $\mathrm{SU}(N)$ Yang-Mills theory at the conformal fixed point, exact hydrodynamic-to-collisionless crossover functions of the SO(8) R-currents can be obtained in the large $N$ limit by applying the anti-de Sitter/conformal field theory (AdS/CFT) correspondence to M theory. In the gravity theory, fluctuating currents are mapped to fluctuating gauge fields in the background of a black hole in $3+1$ dimensional anti-de Sitter space. The electromagnetic self-duality of the $3+1$ dimensional theory implies that the correlators of the R-currents obey a functional constraint similar to that found from particle-vortex duality in $2+1$ dimensional Abelian theories. Thus the $2+1$ dimensional, superconformal Yang Mills theory obeys a “holographic self-duality” in the large $N$ limit, and perhaps more generally.