Membranes with Topological Charge and AdS4/CFT3 Correspondence

TL;DR

This work demonstrates that M-theory on $AdS_4\times Y^7$ with $b_2>0$ hosts topological $U(1)^{b_2}$ gauge symmetries and constructs charged black membranes whose zero-temperature limit yields an $AdS_2\times \mathbb{R}^2\times\text{squashed }Y^7$ throat. The authors develop a universal consistent truncation based on a harmonic two-form on $Y^7$, derive an effective one-dimensional Lagrangian with a topological charge potential $V_Q$ and a scalar potential $V_s$, and solve for both numerical nonzero-temperature backgrounds and analytic extremal solutions. They show wrapped M2-branes do not condense and that the space-time filling M2-brane potential vanishes at $T=0$, indicating stability despite lack of supersymmetry. A rich field theory interpretation is provided in terms of dual quiver Chern–Simons theories with non-diagonal monopole operators and a IIB dual featuring a locally $AdS_3\times$ squashed $Y^7$ geometry tied to an extremal BTZ black hole, suggesting IR quantum critical behavior and a path to understanding the large $N$ entropy.

Abstract

If the second Betti number b_2 of a Sasaki-Einstein manifold Y^7 does not vanish, then M-theory on AdS_4 x Y^7 possesses "topological" U(1)^{b_2} gauge symmetry. The corresponding Abelian gauge fields come from three-form fluctuations with one index in AdS_4 and the other two in Y^7. We find black membrane solutions carrying one of these U(1) charges. In the zero temperature limit, our solutions interpolate between AdS_4 x Y^7 in the UV and AdS_2 x R^2 x squashed Y^7 in the IR. In fact, the AdS_2 x R^2 x squashed Y^7 background is by itself a solution of the supergravity equations of motion. These solutions do not appear to preserve any supersymmetry. We search for their possible instabilities and do not find any. We also discuss the meaning of our charged membrane backgrounds in a dual quiver Chern-Simons gauge theory with a global U(1) charge density. Finally, we present a simple analytic solution which has the same IR but different UV behavior. We reduce this solution to type IIA string theory, and perform T-duality to type IIB. The type IIB metric turns out to be a product of the squashed Y^7 and the extremal BTZ black hole. We discuss an interpretation of this type IIB background in terms of the (1+1)-dimensional CFT on D3-branes partially wrapped over the squashed Y^7.

Figures (7)

• Figure 1: The horizon values of the scalars $\eta_1$, $\eta_2$, and $\chi$ and of the squared Riemann tensor as a function of $T/\mu$. The expected zero-temperature values that follow from \ref{['AttractorSoln']} are indicated by red dashed lines. The fact that none of these quantities diverge as $T \to 0$ shows that the supergravity approximation continues to hold down to arbitrarily small temperatures.
• Figure 2: The dependence of the ratio of entropy density to charge density on $T/\mu$. The dashed line indicates the value $s/\rho = 4 \pi / Q \approx 14.75$ expected from the extremal solution of section \ref{['EXTREMAL']}.
• Figure 3: The dependence of the specific heat at constant chemical potential on $T/\mu$. The dashed line in the plot on the left is a best fit line, showing the linear behavior of the specific heat at low temperatures.
• Figure 4: The dependence of the scalars $\eta_1$, $\eta_2$, and $\chi$ on the radial variable $r$ at zero temperature. We see that the scalars tend to zero at the boundary since our solution asymptotes to $AdS_4 \times Y^7$ in the UV.
• Figure 5: The potential energy per unit area of a probe space-time filling M2-brane as a function of the AdS radial coordinate $r$, at various temperatures. We worked in a gauge where the horizon value of the potential vanishes.