M-theory compactifications to three dimensions with M2-brane potentials

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/Ftheory 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 fourform 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 2W 4 = −W 5 .

The landscape of G-structures in eight-manifold compactifications of M-theory

“…In the case of compactifications down to AdS 3 , σ + (p) can be interpreted [26] as the number of supersymmetries of the background which are preserved by a space-time filling M2-brane placed at p, while σ − (p) counts the number of supersymmetries preserved by a space-time filling M2-antibrane placed at p; these numbers are indicated in the last column of the table.…”

“…Some aspects of N = 2 compactifications of eleven-dimensional supergravity down to AdS 3 were approached in [26] using a nine-dimensional formalism based on the auxiliary 9-manifoldM def.…”

“…The rest of the paper is devoted to the detailed study of the latter case. Section 2 discusses the scalar and one-form bilinears which can be constructed using a basis of K when dim K = 2 and introduces two cosmooth generalized distributions D and D 0 (where D 0 ⊂ D) which 1 Such N = 2 backgrounds were considered in [25] using a nine-dimensional formalism and were also discussed in [26] with similar methods, but without carefully studying the corresponding geometry of the eight-manifold. Certain N = 1 compactifications down to three-dimensional Minkowski space but with torsion-full SU(4) structure were studied in [27, section 3].…”

M-theory on non-Kähler eight-manifolds

“…M-theory compactifications to three dimensions preserving different amounts of supersymmetry have been extensively studied in the literature [10][11][12][13][14][15][16][17]. In references [16,17] a very rigorous and complete study of the geometry of the internal eight-dimensional manifold has been carried out using the theory of codimension-one foliations, which turns out to be the right mathematical tool to characterize it, as suggested in [18].…”

“…Therefore, we cannot perform the conformal transformation that transforms the quadruplet {g, J, ω, Ω} into a Calabi-Yau structure in M 8 , which thus cannot be taken to be a Calabi-Yau four-fold; in particular, the supersymmetry complex spinor is not constant respect to any LeviCivita connection associated to a metric in the conformal class of the physical metric. We can however perform the conformal transformation locally on ever open set U a , and thus we defineg 14) where nowg a andη a are locally defined on U a . The local conformal transformation (3.14) implies, again locally in U a , that…”