We use the GKZ description of periods and certain classes of relative periods on families of Barth-Nieto Calabi-Yau $(l-1)$-folds in order to solve the $l$-loop banana amplitudes with their general mass dependence. As examples we compute the mass dependencies of the banana amplitudes up to the three-loop case and check the results against the known results for special mass values.
We compute numerical approximations of the period integrals for eleven rigid double octic Calabi--Yau threefolds and compare them with the periods of corresponding weight our cusp forms and find, as to be expected, commensurabilities. These give info
rmation on character of the correspondences of these varieties with the associated Kuga-Sato modular threefolds.
We propose machine learning inspired methods for computing numerical Calabi-Yau (Ricci flat Kahler) metrics, and implement them using Tensorflow/Keras. We compare them with previous work, and find that they are far more accurate for manifolds with li
ttle or no symmetry. We also discuss issues such as overparameterization and choice of optimization methods.
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 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 show that Calabi-Yau crystals generate certain Chern-Simons knot invariants, with Lagrangian brane insertions generating the unknot and Hopf link invariants. Further, we make the connection of the crystal brane amplitudes to the topological vertex
formulation explicit and show that the crystal naturally resums the corresponding topological vertex amplitudes. We also discuss the conifold and double wall crystal model in this context. The results suggest that the free energy associated to the crystal brane amplitudes can be simply expressed as a target space Gopakumar-Vafa expansion.
Albrecht Klemm
,Christoph Nega
,Reza Safari
.
(2019)
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"The $l$-loop Banana Amplitude from GKZ Systems and relative Calabi-Yau Periods"
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Christoph Nega
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