We point out that the matrix description of M-theory compactified on Calabi-Yau threefolds is in many respects simpler than the matrix description of a $T^6$ compactification. This is largely because of the differences between D6 branes wrapped on Calabi-Yau threefolds and D6 branes wrapped on six-tori. In particular, if we define the matrix theory following the prescription of Sen and Seiberg, we find that the remaining degrees of freedom are decoupled from gravity.
We derive global constraints on the non-BPS sector of supersymmetric 2d sigma-models whose target space is a Calabi-Yau manifold. When the total Hodge number of the Calabi-Yau threefold is sufficiently large, we show that there must be non-BPS primary states whose total conformal weights are less than 0.656. Moreover, the number of such primary states grows at least linearly in the total Hodge number. We discuss implications of these results for Calabi-Yau geometry.
We use the results of hep-th/0007174 on the simple current classification of open unoriented CFTs to construct half supersymmetry preserving crosscap states for rational Calabi-Yau compactifications. We show that the corresponding orientifold fixed planes obey the BPS-like relation M=exp(i*phi)Q. To prove this relation, it is essential that the worldsheet CFT properly includes the degrees of freedom from the uncompactified space-time component. The BPS-phase phi can be identified with the automorphism type of the crosscap states. To illustrate the method we compute crosscap states in Gepner models with each k_i odd.
We prove that a Kahler supermetric on a supermanifold with one complex fermionic dimension admits a super Ricci-flat supermetric if and only if the bosonic metric has vanishing scalar curvature. As a corollary, it follows that Yaus theorem does not hold for supermanifolds.
We study when Calabi-Yau supermanifolds M(1|2) with one complex bosonic coordinate and two complex fermionic coordinates are super Ricci-flat, and find that if the bosonic manifold is compact, it must have constant scalar curvature.
We show how the smooth geometry of Calabi-Yau manifolds emerges from the thermodynamic limit of the statistical mechanical model of crystal melting defined in our previous paper arXiv:0811.2801. In particular, the thermodynamic partition function of molten crystals is shown to be equal to the classical limit of the partition function of the topological string theory by relating the Ronkin function of the characteristic polynomial of the crystal melting model to the holomorphic 3-form on the corresponding Calabi-Yau manifold.