M-theory compactification, fluxes and AdS_4

TL;DR

This paper derives the complete set of supersymmetry conditions for M-theory compactifications on a seven-dimensional SU(3) structure manifold times a four-dimensional maximally symmetric space, allowing unequal norms of the internal spinors and a possible AdS$_4$ background. It shows that a nonzero 4D cosmological constant requires a nontrivial phase between the internal spinors and constrains fluxes so that the internal vector flux vanishes and the internal space locally splits into circle and six-dimensional half-flat slices, with special local solutions giving warped circle–nearly-Kähler–AdS$_4$ geometries and Calabi–Yau limits. The authors provide explicit singlet-flux solutions, analyze their properties, and examine embeddings into Horava–Witten theory, finding no consistent HW embedding for these singlet-flux cases, while demonstrating that known warped Calabi–Yau HW solutions arise as the perturbative HW limit. Overall, the work clarifies how fluxes and spinor normalizations govern the geometry of M-theory compactifications with SU(3) structure and AdS$_4$ or Minkowski external spaces, highlighting half-flat and nearly-Kähler structures as central features.

Abstract

We analyze supersymmetric solutions of M-theory based an a seven-dimensional internal space with SU(3) structure and a four-dimensional maximally symmetric space. The most general supersymmetry conditions are derived and we show that a non-vanishing cosmological constant requires the norms of the two internal spinors to differ. We find explicit local solutions with singlet flux and the space being a warped product of a circle, a nearly-Kahler manifold and AdS_4. The embedding of solutions into heterotic M-theory is also discussed.