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Register a new userHigher derivatives in Type II and M-theory on Calabi-Yau threefolds
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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.
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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.
We present a list of Calabi-Yau threefolds known to us, and with holonomy groups that are precisely SU(3), rather than a subgroup, with small Hodge numbers, which we understand to be those manifolds with height $(h^{1,1}+h^{2,1})le 24$. With the completion of a project to compute the Hodge numbers of free quotients of complete intersection Calabi-Yau threefolds, most of which were computed in Refs. [1-3] and the remainder in Ref. [4], many new points have been added to the tip of the Hodge plot, updating the reviews by Davies and Candelas in Refs. [1, 5]. In view of this and other recent constructions of Calabi-Yau threefolds with small height, we have produced an updated list.
We investigate the delicate interplay between the types of singular fibers in elliptic fibrations of Calabi-Yau threefolds (used to formulate F-theory) and the matter representation of the associated Lie algebra. The main tool is the analysis and the appropriate interpretation of the anomaly formula for six-dimensional supersymmetric theories. We find that this anomaly formula is geometrically captured by a relation among codimension two cycles on the base of the elliptic fibration, and that this relation holds for elliptic fibrations of any dimension. We introduce a Tate cycle which efficiently describes this relationship, and which is remarkably easy to calculate explicitly from the Weierstrass equation of the fibration. We check the anomaly cancellation formula in a number of situations and show how this formula constrains the geometry (and in particular the Euler characteristic) of the Calabi-Yau threefold.
We investigate the swampland distance conjecture (SDC) in the complex moduli space of type II compactifications on one-parameter Calabi-Yau threefolds. This class of manifolds contains hundreds of examples and, in particular, a subset of 14 geometries with hypergeometric differential Picard-Fuchs operators. Of the four principal types of singularities that can occur - specified by their limiting mixed Hodge structure - only the K-points and the large radius points (or more generally the M-points) are at infinite distance and therefore of interest to the SDC. We argue that the conjecture is fulfilled at the K- and the M-points, including models with several M-points, using explicit calculations in hypergeometric models which contain typical examples of all these degenerations. Together with previous work on the large radius points, this suggests that the SDC is indeed fulfilled for one-parameter Calabi-Yau spaces.
Recent studies (arXiv:1610.07916, arXiv:1711.07921, arXiv:1807.00186) of six-dimensional supersymmetric gauge theories that are engineered by a class of toric Calabi-Yau threefolds $X_{N,M}$, have uncovered a vast web of dualities. In this paper we analyse consequences of these dualities from the perspective of the partition functions $mathcal{Z}_{N,M}$ (or the free energy $mathcal{F}_{N,M}$) of these theories. Focusing on the case $M=1$, we find that the latter is invariant under the group $mathbb{G}(N)times S_N$: here $S_N$ corresponds to the Weyl group of the largest gauge group that can be engineered from $X_{N,1}$ and $mathbb{G}(N)$ is a dihedral group, which acts in an intrinsically non-perturbative fashion and which is of infinite order for $Ngeq 4$. We give an explicit representation of $mathbb{G}(N)$ as a matrix group that is freely generated by two elements which act naturally on a specific basis of the Kahler moduli space of $X_{N,1}$. While we show the invariance of $mathcal{Z}_{N,1}$ under $mathbb{G}(N)times S_N$ in full generality, we provide explicit checks by series expansions of $mathcal{F}_{N,1}$ for a large number of examples. We also comment on the relation of $mathbb{G}(N)$ to the modular group that arises due to the geometry of $X_{N,1}$ as a double elliptic fibration, as well as T-duality of Little String Theories that are constructed from $X_{N,1}$.
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