Abstract
We study a class of compactifications of M-theory to three dimensions that preserve N=2 supersymmetry and which have the defining feature that a probe space-time filling M2 brane feels a non-trivial potential on the internal manifold. Using M-theory/F-theory duality such compactifications include the uplifts of 4-dimensional N=1 type IIB compactifications with D3 potentials to strong coupling. We study the most general 8-dimensional manifolds supporting these properties, derive the most general flux that induces an M2 potential, and show that it is parameterised in terms of two real vectors. We study the supersymmetry equations when only this flux is present and show that over the locus where the M2 potential is non-vanishing the background takes the form of a Calabi-Yau three-fold fibered over a 2-dimensional base spanned by the flux vectors, while at the minima of the potential the flux vanishes. Allowing also for non-vanishing four-form flux with one leg in the internal directions we find that the Calabi-Yau three-fold in the fibration is replaced by an SU(3)-structure manifold with torsion classes satisfying 2 W_4=-W_5.
Highlights
We study a class of compactifications of M-theory to three dimensions that preserve N = 2 supersymmetry and which have the defining feature that a probe space-time filling M2 brane feels a non-trivial potential on the internal manifold
In this paper we studied N = 2 compactifications of M-theory to three dimensions which have the defining property that a potential is induced for probe space-time filling M2branes
Such backgrounds are relevant for many model building applications, ranging from moduli stabilisation to flavour physics, all of which rely on backgrounds that involve potentials for space-time filling probe D3-branes on the F-theory side, which by the F-theory/M-theory duality implies an M2-potential
Summary
In the case that all Majorana-Weyl components are everywhere non-vanishing we are dealing with an 8-dimensional manifold with SU(3) structure which preserves (a maximum of) N = 4 supersymmetry in 3 dimensions. This spinor is obviously orthogonal to ξ1,2 and because the matrix γαβ is antisymmetric in its spinorial indices, it is orthogonal to χ1 In this way we have 4 non-vanishing Majorana-Weyl spinors, two of positive and two of negative chirality, which define a SU(3) structure in 8 dimensions. At generic values of this parameter, all the three vectors are independent and of non-vanishing norm This is the case of a local SU(3) structure. It should be clear that following a suitable normalisation the forms J and Ω (or its real components φ and ρ) can be seen as the forms defining an SU(3)-structure on the space orthogonal to the vectors V± and V3.
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