Supersymmetry, G-flux and Spin(7) manifolds
Bobby Acharya, Sergei Gukov, Xenia de la Ossa
TL;DR
This work analyzes M-theory compactifications on 8-manifolds with Spin(7) holonomy in the presence of background 4-form flux, showing that supersymmetry allows nonzero G-flux when it lies in the $\mathbf{27}^+$ representation of $Spin(7)$. It identifies a leading 3D $\mathcal{N}=1$ superpotential $W=\frac{1}{2\pi}\int_X G\wedge\Phi$, matches 11D SUSY conditions to the 3D EFT, and discusses perturbative and nonperturbative corrections to the potential, as well as Chern-Simons masses for Abelian vectors. The analysis provides precise cohomological constraints on allowable fluxes (notably $H^4_{\mathbf{7}^+}=0$, $G\in H^4_{\mathbf{27}^+}$) and highlights the role of M2/M5 instantons in lifting flat directions, with implications for the structure of 3D vacua in $M$-theory on Spin(7) manifolds.
Abstract
In this note we study warped compactifications of M-theory on manifolds of Spin(7) holonomy in the presence of background 4-form flux. The explicit form of the superpotential can be given in terms of the self -dual Cayley calibration on the Spin(7) manifold, in agreement with the general formula propsed in hep-th/9911011.