M-theory on Calabi-Yau Five-Folds

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 Kahler and complex structure moduli of the five-fold reside in 2a and 2b super-multiplets, respectively. Further 2b 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 show that the scalar potential from (2,2) flux is consistent with one-dimensional N=2 supersymmetry and can be described in terms of a superpotential. This superpotential can also be obtained from a Gukov-type 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.