Moduli-dependent Calabi-Yau and SU(3)-structure metrics from Machine Learning
Calabi-Yau manifolds play a crucial role in string compactifications. Yau's theorem guarantees the existence of a metric that satisfies the string's equation of motion. However, Yau's proof is non-constructive, and no analytic expressions for metrics on Calabi-Yau threefolds are known. We use machine learning, more precisely neural networks, to learn Calabi-Yau metrics and their complex structure moduli dependence. I will start with an introduction to Calabi-Yau manifolds and their moduli. After that, I will give a brief introduction to neural networks. Using an example, I will then illustrate how we train neural networks to find Calabi-Yau metrics by solving the underlying partial differential equations. The approach generalizes to more general manifolds and can hence also be used for manifolds with reduced structure, such as SU(3) structure or G2 manifolds, which feature in string compactifications with flux and in the M-theory formulation of string theory, respectively. I will illustrate this generalization for a particular SU(3) structure metric and compare the machine learning result to the known, analytic expression.