Generating Special Triangulations with Transformers

Triangulations of 4D reflexive polytopes are essential for constructing Calabi-Yau threefolds in string theory, but their combinatorial complexity makes them difficult to compute using traditional methods. The authors investigate whether transformer-based machine learning models can learn to generate these structures, potentially overcoming the limitations of human intuition and classical algorithms.

The authors propose the CYTransformer, an encoder-decoder model designed to map a sequence of polytope vertices to a sequence of simplices representing a valid FRST. To handle the geometric nature of the data, they developed a custom tokenization scheme that maps simplices to discrete tokens. The training process incorporates data augmentation to ensure invariance under vertex and simplex permutations. Furthermore, the authors implement a self-improvement loop where the model generates new candidate triangulations, validates them using external software (CYTools), and retrains on the successful outputs to expand its knowledge base.

The study demonstrates that transformers can effectively learn the distribution of FRSTs, a task that is non-trivial due to the strict geometric constraints and the scarcity of valid triangulations within the space of all possible simplex combinations. The models not only generate valid FRSTs for polytopes seen during training but also generalize to unseen polytopes. The self-improvement strategy proves effective, allowing the model to enhance its performance beyond the limitations of the initial training data. These results suggest that transformers can successfully navigate complex, high-dimensional combinatorial spaces relevant to theoretical physics.

This work provides a scalable, machine-learning-based alternative to traditional algorithms for exploring the string theory landscape. By automating the generation of FRSTs, the CYTransformer facilitates the classification of Calabi-Yau manifolds and offers a new methodology for tackling complex geometric and combinatorial problems in mathematics and physics that were previously computationally prohibitive.