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Fluxes in M-theory on 7-manifolds: G-structures and Superpotential

Klaus Behrndt, Claus Jeschek

TL;DR

This work analyzes M-theory compactified on 7-manifolds with 4-form fluxes that preserve four supercharges, focusing on the case of two SU(3) singlet spinors to realize an SU(3) structure on a six-dimensional base X_6. By decomposing the 11d spinor, flux, and geometry, the authors derive SUSY (BPS) constraints that relate the warp factor, the Freud–Rubin parameter, and flux components organized into G-structure representations; they show that, in the one-spinor case, internal fluxes vanish and the cosmological constant is set by m, while in the two-spinor case, m must vanish and the 4d superpotential is determined by flux projections onto X_6, with precise SU(3) decompositions for H and G. The analysis yields a holomorphic W in terms of projected fluxes and links the warp-factor behavior to nonprimitive (2,1) flux components, providing a geometric realization of the 4d effective theory and clarifying connections to IIA reductions, MQCD setups, and domain-wall interpretations. These results illuminate how fluxes and G-structures stabilize moduli and generate a nontrivial 4d potential in M-theory compactifications, with explicit conditions for when X_6 is complex or when SUSY vacua reduce to simpler geometries.

Abstract

We consider compactifications of M-theory on 7-manifolds in the presence of 4-form fluxes, which leave at least four supercharges unbroken. We focus especially on the case, where the 7-manifold supports two spinors which are SU(3) singlets and the fluxes appear as specific SU(3) structures. We derive the constraints on the fluxes imposed by supersymmetry and calculate the resulting 4-dimensional superpotential.

Fluxes in M-theory on 7-manifolds: G-structures and Superpotential

TL;DR

This work analyzes M-theory compactified on 7-manifolds with 4-form fluxes that preserve four supercharges, focusing on the case of two SU(3) singlet spinors to realize an SU(3) structure on a six-dimensional base X_6. By decomposing the 11d spinor, flux, and geometry, the authors derive SUSY (BPS) constraints that relate the warp factor, the Freud–Rubin parameter, and flux components organized into G-structure representations; they show that, in the one-spinor case, internal fluxes vanish and the cosmological constant is set by m, while in the two-spinor case, m must vanish and the 4d superpotential is determined by flux projections onto X_6, with precise SU(3) decompositions for H and G. The analysis yields a holomorphic W in terms of projected fluxes and links the warp-factor behavior to nonprimitive (2,1) flux components, providing a geometric realization of the 4d effective theory and clarifying connections to IIA reductions, MQCD setups, and domain-wall interpretations. These results illuminate how fluxes and G-structures stabilize moduli and generate a nontrivial 4d potential in M-theory compactifications, with explicit conditions for when X_6 is complex or when SUSY vacua reduce to simpler geometries.

Abstract

We consider compactifications of M-theory on 7-manifolds in the presence of 4-form fluxes, which leave at least four supercharges unbroken. We focus especially on the case, where the 7-manifold supports two spinors which are SU(3) singlets and the fluxes appear as specific SU(3) structures. We derive the constraints on the fluxes imposed by supersymmetry and calculate the resulting 4-dimensional superpotential.

Paper Structure

This paper contains 9 sections, 92 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Warp compactification in the presence of fluxes
  3. Decomposition of the Killing spinor
  4. Back reaction onto the geometry
  5. BPS constraints
  6. Manifolds with a single 7-spinor
  7. Manifolds with two 7-spinors
  8. Discussion
  9. Examples

Figures (1)

  • Figure 1: We have shown here a simple case, where the vector $v$ built out of two 7-d spinors gives a foliation of the 7-manifold $X_7$ by a 6-manifold $X_6$. This foliation is not unique since on any 7-manifold that is spin exists at least three globally well-defined vectors.