M-theory on Calabi-Yau Five-Folds

Abstract

We study the compactification of M-theory on Calabi-Yau five-folds and the resulting N = 2 super-mechanics theories. By explicit reduction from 11 dimensions, including both bosonic and fermionic terms, we calculate the one-dimensional effective action and show that it can be derived from an N = 2 super-space action. We find that the Kähler and complex structure moduli of the five-fold reside in 2a and 2b supermultiplets, respectively. Constrained 2a super-multiplets arise from zero-modes of the M-theory three-form and lead to cross-couplings between 2a and 2b multiplets. Fermionic zero modes which arise from the (1, 3) sector of the 11-dimensional gravitino do not have bosonic super-partners and have to be described by purely fermionic super-multiplets in one dimension. We also study the inclusion of flux and discuss the consistency of the scalar potential with one-dimensional N = 2 supersymmetry and how it can be described in terms of a superpotential. This superpotential can also be obtained from a Gukovtype formula which we present. Supersymmetric vacua, obtained by solving the F-term equations, always have vanishing vacuum energy due to the form of this scalar potential. We show that such supersymmetric solutions exist for particular examples. Two substantial appendices develop the topology and geometry of Calabi-Yau five-folds and the structure of one-dimensional N = 2 supersymmetry and supergravity to the level of generality required for our purposes.Keywords: M-Theory, Flux compactifications, Field Theories in Lower Dimensions, Superspaces. 65 C.2 Local N = 2 supersymmetry 69 -1 -will demonstrate that this can indeed frequently be achieved. In particular, we show that the consistency condition can be satisfied for the Calabi-Yau five-fold defined by the zero locus of a septic polynomial in P 6 . The "septic" is arguably the simplest five-fold and the analogue of the quintic three-fold in P 4 . The one-dimensional effective action will be calculated as an expansion in powers of β. As a first step we consider the situation at zeroth order in β. Effects from flux or membranes only come in at order β and are, therefore, not relevant at this stage. In particular, we clarify the relation between Calabi-Yau topology/geometry and the structure of the onedimensional supermechanics induced by M-theory at this lowest order in β. Many aspects of this relation are analogous to what happens for compactifications on lower-dimensional Calabi-Yau manifolds, others, as we will see, are perhaps less expected. The topology of a Calabi-Yau five-fold X is characterised by six a priori independent Hodge numbers, namely h 1,1 (X), h 1,2 (X), h 1,3 (X), h 2,2 (X), h 1,4 (X) and h 2,3 (X). In analogy with the four-fold case [15], an index theorem calculation together with the Calabi-Yau condition c 1 (X) = 0, leads to one relation between those six numbers. The moduli space of a Calabi-Yau manifold consists (locally) of a direct product of a Kähler and a complex structure moduli space [16]. For Calabi-Yau five-folds, these two par...

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