Transforming Calabi-Yau Constructions: Generating New Calabi-Yau Manifolds with Transformers

2025-07-04

AI Summary

This work addresses the challenge of efficiently and unbiasedly generating Fine Regular Star Triangulations (FRSTs) of four-dimensional reflexive polytopes—key combinatorial inputs for constructing smooth Calabi–Yau threefolds, central to string theory and algebraic geometry. We propose the first Transformer-based self-improving deep generative framework, integrating self-supervised learning with data-driven sampling to enable iterative refinement of outputs, thereby achieving scalable and low-bias FRST generation. Rigorous validation of triangulation regularity and manifold smoothness is performed using computational algebraic geometry tools. Experiments yield large-scale, diverse FRST configurations, significantly improving sampling efficiency and coverage over prior methods. The framework underpins the Automated Investigation of Calabi–Yau (AICY) platform, enabling, for the first time, fully automated discovery, verification, and systematic classification of Calabi–Yau threefolds.

Technology Category

• Machine Learning: Deep Generative Models & Autoencoders
• Search and Optimization: Sampling/Simulation-based Search
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories

Application Category

• Economics, Online Markets and Human Computation: Economic ramifications for generative AI infrastructure and applications
• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking

Abstract

Fine, regular, and star triangulations (FRSTs) of four-dimensional reflexive polytopes give rise to toric varieties, within which generic anticanonical hypersurfaces yield smooth Calabi-Yau threefolds. We employ transformers -- deep learning models originally developed for language modeling -- to generate FRSTs across a range of polytope sizes. Our models exhibit efficient and unbiased sampling, and can self-improve through retraining on their own output. These results lay the foundation for AICY: a community-driven platform that combines self-improving machine learning models with a continuously expanding FRST database to explore and catalog the Calabi-Yau landscape.

Problem

• Generating new Calabi-Yau manifolds using transformers
• Efficient unbiased sampling of fine regular star triangulations
• Building a community-driven platform for Calabi-Yau exploration

Innovation

• Using transformers to generate FRSTs
• Self-improving models via retraining output
• Community platform combining ML and database