M-theory on eight-manifolds revisited: N=1 supersymmetry and generalized Spin(7) structures

TL;DR

The paper analyzes ${\cal N}=1$ M-theory vacua on warped products ${\cal M}\times_w X$ with $X$ eight-dimensional, showing that a nowhere-vanishing spinor on $X$ lifts to a Spin(7) reduction on the nine-manifold $Y=X\times S^1$ and motivates a generalized Spin(7) description on $TY\oplus T^*Y$. It derives SUSY conditions as differential equations for Spin(7) bispinors, expresses the intrinsic torsion and flux components in terms of the warp factor $\Delta$ and a bilinear one-form $L$, and identifies two differential constraints that $L$ must satisfy. A small-flux expansion demonstrates how, near Spin(7) holonomy, fluxes and torsion are controlled by $L$ and $\Delta$, with explicit behavior in noncompact and AdS/Minkowski cases. The generalized Spin(7) framework is extended to a seven-dimensional reduction and related to generalized $G_2$ structures, highlighting a coherent geometric language for eight-manifold compactifications and small-flux perturbations. The work opens paths to explicit global examples, Hitchin-type functionals in this setting, and KK reductions to three dimensions.

Abstract

The requirement of ${\cal N}=1$ supersymmetry for M-theory backgrounds of the form of a warped product ${\cal M}\times_{w}X$, where $X$ is an eight-manifold and ${\cal M}$ is three-dimensional Minkowski or AdS space, implies the existence of a nowhere-vanishing Majorana spinor $ξ$ on $X$. $ξ$ lifts to a nowhere-vanishing spinor on the auxiliary nine-manifold $Y:=X\times S^1$, where $S^1$ is a circle of constant radius, implying the reduction of the structure group of $Y$ to $Spin(7)$. In general, however, there is no reduction of the structure group of $X$ itself. This situation can be described in the language of generalized $Spin(7)$ structures, defined in terms of certain spinors of $Spin(TY\oplus T^*Y)$. We express the condition for ${\cal N}=1$ supersymmetry in terms of differential equations for these spinors. In an equivalent formulation, working locally in the vicinity of any point in $X$ in terms of a `preferred' $Spin(7)$ structure, we show that the requirement of ${\cal N}=1$ supersymmetry amounts to solving for the intrinsic torsion and all irreducible flux components, except for the one lying in the $\bf{27}$ of $Spin(7)$, in terms of the warp factor and a one-form $L$ on $X$ (not necessarily nowhere-vanishing) constructed as a $ξ$ bilinear; in addition, $L$ is constrained to satisfy a pair of differential equations. The formalism based on the group $Spin(7)$ is the most suitable language in which to describe supersymmetric compactifications on eight-manifolds of $Spin(7)$ structure, and/or small-flux perturbations around supersymmetric compactifications on manifolds of $Spin(7)$ holonomy.