Effervescent Spikes in M-theory

TL;DR

The paper demonstrates that AdS$_3\times$S$^3\times$S$^3$ solutions warped over a Riemann surface $\Sigma$ with superconformal symmetry $D(2,1;\gamma)\oplus D(2,1;\gamma)$ arise as near-horizon scaling limits of a single system of M2–M5–M5' spikes in M-theory, with the parameter $\gamma$ selecting how the three radial directions are scaled. Negative $\gamma$ yields smooth bubbling geometries, while positive $\gamma$ leads to limits where bubbles collapse into singular brane sources; the sign of $\gamma$ also dictates which AdS asymptotics emerge (AdS$_7\times S^4$ vs AdS$_4\times S^7$). The authors construct the 1/4-BPS M2–M5–M5' intersections, map AdS$_3$ solutions to these brane configurations using scale-invariant coordinates, and analyze M2 charges, brane probes, and geometric transitions (the mohawk) that produce bubbling AdS spaces. They establish that bubbling negative-$\gamma$ solutions originate from geometric transitions of M2–M5 spikes and show the universality of the Poincaré supersymmetries across the family, with the superconformal extension encoded in the scaling. The work unifies the AdS$_3$ solutions as a one-parameter family of near-brane limits and suggests avenues for extending these constructions to more general Janus-type geometries and higher-BPS configurations.

Abstract

AdS$_3 \times$ S$^3 \times$ S$^3$ solutions warped over a Riemann surface, $Σ$, are indexed by a parameter, $γ$, that defines the superconformal algebra, $D(2,1; γ) \oplus D(2,1; γ)$ they preserve. We show that these solutions come from multiple back-reacted M2-M5 spikes, and that different values of $γ$ correspond to different scaling limits of the same M2-M5 solutions. We find that when $γ$ switches from positive to negative, the infrared region of the AdS$_3$ switches from the tip of spikes, far from the M5 branes, to the bottom of the spikes, far from the M2 branes. We also explain how the bubbling negative-$γ$ solutions emerge from the geometric transition of multiple M2-M5 spikes.

Figures (8)

• Figure 1: The Poincaré upper half-plane, showing the boundary values of $G$ with an even number of flip points at finite $\xi_j$ (and hence no flip at infinity). Path $B$ defines a homology sphere, ${S}^4$, while paths $A$ and $C$ define homology spheres, ${S'}^4$. There is a net M5 charge at infinity and so the cycle $A+C$ is not contractible, and the cycles $A$ and $C$ carry independent 5-brane charges.
• Figure 2: The Poincaré upper half-plane, showing the boundary values of $G$ with an odd number of flip points at finite $\xi_j$ (and a flip at infinity). The paths $A$ and $C$ define homology spheres, ${S'}^4$, while $B$ and $D$ define homology spheres, ${S}^4$. At infinity there is an (AdS$_4/\mathbb{Z}_2) \times$S$^7$, and the M2 charge is equal to the integral of $F_7$ on this S$^7$ (see Section \ref{['sec:M2chg-spikes']}).
• Figure 3: A plot of the steepness functions, $u^2 z$ (in red, bottom-left to top-right) and $v^2 w$ (in blue, top-left to bottom-right) for $\xi_j =(-5, -3,-1,1,3,5)$. The function, $\Phi$, involves a constant of integration and so the actual steepness of the brane spikes can involve a vertical translation of this figure. Note that the plateaus and linear behavior alternate between the two functions.
• Figure 4: Deforming the cycles of Fig. \ref{['fig:Topology2']}, and introducing cycles $E$ and $F$ around $\xi_5$ and $\xi_1$, respectively. The cycles $A,B,C,D$ are now to be viewed as $7$-cycles with topology $S^4 \times S^3$, while $E$ and $F$ are to be viewed as $7$-cycles with topology $S^7$. The cycles $A+C+E$ and $B+D+F$ are homologous and can be deformed to wrap the $S^7$ at infinity.
• Figure 5: An M2-M5 probe at $\xi_0$ (denoted by $\times$ in the top panel) has exactly the same M2-M5 charge ratio, $\frac{Q_{M2}}{Q_{M5}}$, as a tiny blue interval between $\xi_{-1}$ and $\xi_0$ in the bottom panel.