Abstract
Hodge numbers of Calabi-Yau manifolds depend non-trivially on the underlying manifold data and they present an interesting challenge for machine learning. In this letter we consider the data set of complete intersection Calabi-Yau four-folds, a set of about 900,000 topological types, and study supervised learning of the Hodge numbers h1,1 and h3,1 for these manifolds. We find that h1,1 can be successfully learned (to 96% precision) by fully connected classifier and regressor networks. While both types of networks fail for h3,1, we show that a more complicated two-branch network, combined with feature enhancement, can act as an efficient regressor (to 98% precision) for h3,1, at least for a subset of the data. This hints at the existence of an, as yet unknown, formula for Hodge numbers.
Highlights
Topological quantities of manifolds, such as Betti or Hodge numbers, are often non-trivially related to the data describing the underlying manifold and tend to be difficult to work out
We will address this problem for complete intersection Calabi-Yau (CICY) four-folds and their Hodge numbers
The complete set of CICY three-folds was the first large dataset of Calabi-Yau manifolds to be constructed [2,3]. It consists of 7890 different topological types of manifolds which have provided string theorists and mathematicians alike with a fertile ground for exploration
Summary
Topological quantities of manifolds, such as Betti or Hodge numbers, are often non-trivially related to the data describing the underlying manifold and tend to be difficult to work out. Explicit formulae are usually not known and calculations rely on complicated and frequently computationally intense algorithms (see, for example, the volume [1] and references therein for applications of computational algebraic geometry to string and gauge theories) For this reason, such topological properties are an interesting and challenging playground for machine learning. The complete set of CICY three-folds was the first large dataset of Calabi-Yau manifolds to be constructed [2,3] It consists of 7890 different topological types of manifolds which have provided string theorists and mathematicians alike with a fertile ground for exploration We will explore, within the context of supervised learning, if and to what extent Hodge numbers of CICY four-folds can be learned by neural networks
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