Supermembranes with fewer supersymmetries

TL;DR

Duff, Lü, Pope, and Sezgin extend the D=11 supermembrane by replacing the traditional round S^7 with any seven-dimensional Einstein space M^7, yielding new p-brane solutions whose symmetry is $P_3\times G$ and which preserve $N/16$ supersymmetry where $N$ is the number of Killing spinors on $M^7$. The preserved supersymmetry depends on the orientation (skew-whiffing) and ranges from $1/16$ to none, while all solutions saturate the Bogomol'nyi bound with mass $m$ equal to the Page charge $Z$ and to the parameter $k$. The paper analyzes zero modes, includes source terms, and shows that simultaneous dimensional reduction on $S^1$ produces new $D=10$ elementary strings with reduced supersymmetries; it also explores Ricci-flat $M^7$ cases (e.g., Joyce manifolds with $G_2$ holonomy) where the SUSY counting changes and yields distinct zero-mode structures. These results reveal a broader spectrum of supersymmetric branes and deepen the connections between geometry (Killing spinors on Einstein spaces) and nonperturbative objects in M-theory and string theory.

Abstract

The usual supermembrane solution of $D=11$ supergravity interpolates between $R^{11}$ and $AdS_4 \times round~S^7$, has symmetry $P_3 \times SO(8)$ and preserves $1/2$ of the spacetime supersymmetries for either orientation of the round $S^7$. Here we show that more general supermembrane solutions may be obtained by replacing the round $S^7$ by any seven-dimensional Einstein space $M^7$. These have symmetry $P_3 \times G$, where $G$ is the isometry group of $M^7$. For example, $G=SO(5) \times SO(3)$ for the squashed $S^7$. For one orientation of $M^7$, they preserve $N/16$ spacetime supersymmetries where $1\leq N \leq 8$ is the number of Killing spinors on $M^7$; for the opposite orientation they preserve no supersymmetries since then $M^7$ has no Killing spinors. For example $N=1$ for the left-squashed $S^7$ owing to its $G_2$ Weyl holonomy, whereas $N=0$ for the right-squashed $S^7$. All these solutions saturate the same Bogomol'nyi bound between the mass and charge. Similar replacements of $S^{D-p-2}$ by Einstein spaces $M^{D-p-2}$ yield new super $p$-brane solutions in other spacetime dimensions $D\leq 11$. In particular, simultaneous dimensional reduction of the above $D=11$ supermembranes on $S^1$ leads to a new class of $D=10$ elementary string solutions which also have fewer supersymmetries.