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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.
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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 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.
The four- and five-dimensional effective actions of Calabi-Yau threefold compactifications are derived with a focus on terms involving up to four space-time derivatives. The starting points for these reductions are the ten- and eleven-dimensional supergravity actions supplemented with the known eight-derivative corrections that have been inferred from Type II string amplitudes. The corrected background solutions are determined and the fluctuations of the Kahler structure of the compact space and the form-field background are discussed. It is concluded that the two-derivative effective actions for these fluctuations only takes the expected supergravity form if certain additional ten- and eleven-dimensional higher-derivative terms for the form-fields are included. The main results on the four-derivative terms include a detailed treatment of higher-derivative gravity coupled to Kahler structure deformations. This is supplemented by a derivation of the vector sector in reductions to five dimensions. While the general result is only given as an expansion in the fluctuations, a complete treatment of the one-Kahler modulus case is presented for both Type II theories and M-theory.
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.
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.
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