[Paper Review] The Calabi-Yau Landscape: from Geometry, to Physics, to Machine-Learning

Yang‐Hui He|arXiv (Cornell University)|Dec 7, 2018

Black Holes and Theoretical PhysicsPhysics and Astronomy291 references46 citations

TL;DR

A survey and pedagogical exploration of Calabi-Yau manifolds across geometry, physics, and data-driven machine learning, highlighting datasets (e.g., CICY, Kreuzer–Skarke) and early ML applications to algebraic geometry.

ABSTRACT

We present a pedagogical introduction to the recent advances in the computational geometry, physical implications, and data science of Calabi-Yau manifolds. Aimed at the beginning research student and using Calabi-Yau spaces as an exciting play-ground, we intend to teach some mathematics to the budding physicist, some physics to the budding mathematician, and some machine-learning to both. Based on various lecture series, colloquia and seminars given by the author in the past year, this writing is a very preliminary draft of a book to appear with Springer, by whose kind permission we post to ArXiv for comments and suggestions.

Motivation & Objective

• Introduce the mathematical and physical motivation for Calabi–Yau manifolds and their role in string theory.
• Summarize major Calabi–Yau datasets and their topological/geometric quantities.
• Explain how computational tools and data-driven methods are used to study Calabi–Yau spaces and their properties.
• Highlight recent developments at the intersection of machine learning and algebraic geometry with Calabi–Yau data.

Proposed method

• Review foundational concepts in Calabi–Yau geometry (Ricci-flat Kähler metrics, Calabi conjecture and Yau’s theorem).
• Describe key Calabi–Yau datasets (CICY, KS data, hypersurfaces in weighted projective spaces) and their topological statistics.
• Discuss computational algebraic geometry tools (e.g., Gröbner bases, exact sequences) and databases used to compute invariants (Hodge numbers, cohomology).
• Introduce machine-learning paradigms applied to algebraic geometry and Calabi–Yau data (regression, neural networks, data-driven pattern discovery).
• Map the landscape of compact and non-compact CY varieties and their physical interpretations (compactification, quiver gauge theories, brane tilings).

Experimental results

Research questions

• RQ1 What are the main data-driven patterns that can be learned from Calabi–Yau datasets?
• RQ2 How can machine learning assist in computing or predicting topological and geometric invariants of Calabi–Yau manifolds?
• RQ3 What databases exist for Calabi–Yau manifolds and what topological/statistical quantities do they encode?
• RQ4 How do computational tools integrate with physics-inspired questions in string theory and algebraic geometry?
• RQ5 What is the role of ML in extending or discovering new Calabi–Yau structures and their properties?

Key findings

• Calabi–Yau manifolds serve as a central intersection of geometry, physics, computation, and data science.
• Datasets such as CICY and Kreuzer–Skarke provide large-scale catalogs (and substantial numbers) of Calabi–Yau spaces and invariants.
• Machine-learning approaches have shown promise in learning algebraic geometry data, including cohomology computations and other invariants, on CY datasets.
• A broad ecosystem of computer algebra systems and databases (Macaulay2, Singular, Bertini, GAP, MAGMA, PARI/GP, SageMath) supports CY research and data mining.
• The work emphasizes a data-driven, computational approach to CY landscapes, including both compact and non-compact varieties, and cross-disciplinary connections to brane tilings, quiver representations, and AdS/CFT.
• The author frames the CY landscape as a playground for exploring computational algebraic geometry and data science rather than as a traditional theory-focused text.