Title:

Computing Ricci-flat Metrics for Highly Deformed Calabi-Yau Manifolds

String theory aims to unify gravity, electromagnetism, and the weak and strong nuclear forces in one theory, arguing that in addition to the four known dimensions, there are six additional dimensions. Curled-up six-dimensional Calabi-Yau (CY) manifolds show the geometry of the extra dimensions, and they have a strong effect on the properties of the observable four-dimensional universe. Computing the Ricci-flat metric with the complex Monge-Ampere equation is necessary to determine whether a given CY geometry is a realistic model for our universe, and machine learning techniques are very useful to solve the complex Monge-Ampère equation because there is no analytical solution. The SYZ conjecture states that in highly deformed Calabi-Yau manifolds (as the parameter ψ approaches infinity), the metric for the complex Monge-Ampère equation also satisfies the real Monge-Ampère equation on the CY manifold, which can be verified by solving for both the real and complex Monge-Ampère equation on the manifold and comparing the accuracy of the machine learning model. Previous work has focused on using Multi-Layer Perceptrons, a type of neural network with weights on the nodes and activation functions. We used TensorFlow Dense networks and implemented adaptive Fourier features, which capture high frequency areas of the CY manifold, particularly in deformed manifolds such as those studied in the large ψ limit.

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