Theoretical Physics Master Thesis: Zack Kite

  • Date: –11:00
  • Location:
  • Lecturer: Zack Kite
  • Organiser: Magdalena Larfors
  • Contact person: Magdalena Larfors
  • Seminarium

Calabi-Yau Metrics using Numerical Methods and Multilayer perceptrons

We study and implement a method first suggested by Yau and later developed by Donaldson which provides a means of numerically approximating the Ricci-flat Kähler metric on a Calabi-Yau manifold. This method involves finding the balanced metric on an embedding of the manifold in complex projective space, an embedding which is given by the sections of a holomorphic line bundle. We implement the algorithm in the context of the Fermat Quintic threefold. Using the $\sigma$-measure we evaluated how close the approximate Kähler volume form is to the Ricci-flat volume form. As noted by Ashmore et. al. the computational time efficiency of the Donaldson’s algorithm grows factorially with the order of the monomial line bundle sections. This presents issues when considering other more complex Calabi-Yau manifolds. For this reason, there have been efforts to apply machine learning to this and other such problems in computational geometry with the intention of bypassing computational expensive algorithms like that of Donaldson’s. Identifying the $\sigma$-measure as an objective function we recast the problem of finding the numeric Calabi-Yau metric as an optimisation problem. Upon constructing a feedforward multi-layer perceptron (MLP) we use a process known as stochastic gradient descent to find the local minima of the $\sigma$-measure. This is done with partial success yielding a MLP model capable of approximating the metric to an error on par with the “first-order” approximation given by the Donaldson’s algorithm. Finally, we discuss potential ways of improving the predictive performance of the MLP such as alternative loss functions as well as Bayesian hyperparameter optimisation.