Quantum critical transport, duality, and M-theory

TL;DR

The paper investigates finite-temperature charge transport in 2+1D CFTs through two complementary routes. In the Abelian sector, duality in the easy-plane ${ m CP}^1$ model imposes functional constraints among longitudinal and transverse current correlators, constraining the possible forms of the hydrodynamic-to-collisionless crossover. In the M2-brane theory, holography reveals an electromagnetic self-duality in the AdS$_4$ bulk that enforces a product relation between current self-energies, yielding a frequency-independent conductivity at zero momentum and a rich hydrodynamic-to-collisionless crossover at finite momentum. The work highlights a holographic mechanism for non-Abelian duality in 2+1D CFTs and contrasts it with cases (like D2-branes) where the duality is broken by a dilaton. Overall, it uncovers universal constraints on finite-temperature transport arising from dualities in higher-dimensional holographic descriptions.

Abstract

We consider charge transport properties of 2+1 dimensional conformal field theories at non-zero 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 the n=8 supersymmetric, 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 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.

Figures (4)

• Figure 1: Phase diagram mv of the easy-plane non-compact ${\mathbb C}{\mathbb P}^1$ model (Eq. (\ref{['sz']})) in 2 spatial dimensions as a function of the coupling $s$ and temperature $T$. The quantum critical point is at $s=s_c$, $T=0$. The finite $T$ correlations of the CFT describe the shaded quantum critical region; the boundary of the shaded region is a crossover into a different physical region, not a phase transition. The full lines are Kosterlitz-Thouless (KT) phase transitions. The KT line for $s<s_c$ describes the disappearance of quasi-long-range $xy$ order of $\vec{N}$. The KT transition for $s>s_c$ describes the deconfinement of $z$ quanta which are logarithmically bound by the Coulomb interaction in the low temperature phase into particle-anti-particle pairs. The phase diagram can also be described in terms of the dual $w$ theory in Eq. (\ref{['sw']}). Duality interchanges the two sides of $s=s_c$ ($T$ remains invariant under duality), and the $z$ Coulomb phase is interpreted as a $w$ Higgs phase and vice versa.
• Figure 2: Imaginary part of the retarded function $C_{yy}(\omega,k)$, plotted in units of $(-\chi)$, as a function of dimensionless frequency $w\equiv 3\omega/(4\pi T)$, for several values of dimensionless momentum $q\equiv 3k/(4\pi T)$. Curves from left to right correspond to $q=0,0.5,1.0,2.0,3.0$. Left: $\mathop{\mathrm{Im}} C_{yy}(w,q)$, Right: $\mathop{\mathrm{Im}} C_{yy}(w,q)/w$.
• Figure 3: Imaginary part of the retarded function $C_{tt}(w,q)/q^2$, plotted in units of $(-\chi)$, as a function of dimensionless frequency $w\equiv 3\omega/(4\pi T)$, for several values of dimensionless momentum $q\equiv 3k/(4\pi T)$. Curves from left to right correspond to $q=0.2,0.5,1.0$ (left panel), and $q=1.0,2.0,3.0,4.0$ (right panel). The dashed curves are plots of Eq. (\ref{['chigh']}) divided by $k^2$.
• Figure 4: The position of the peak of the spectral function in Fig. \ref{['fig:ImCtt']}. The dashed line is $w=q$.