Exploring Line Bundle Standard Models with Transformers

2026-06-30

AI Summary

This work addresses the challenge of searching for line bundle standard models in heterotic string compactifications, where the solution space is high-dimensional and discrete, subject to multiple intricate physical constraints. The authors propose LB-Explorer, the first framework to integrate Transformer-based reinforcement learning into string landscape exploration. It automatically searches for line bundle configurations on smooth CICY threefolds that realize an SU(5) gauge group and satisfy its breaking conditions. By incorporating a CP-SAT solver, LB-Explorer forms a neuro-symbolic hybrid system that precisely enforces core physical constraints—including anomaly cancellation, supersymmetry preservation, chiral spectrum requirements, and the absence of exotic matter. This approach enables end-to-end scalable solving and efficiently generates a large number of viable candidate models. The implementation is publicly released.

Abstract

We propose a Transformer-based Reinforcement Learning architecture,"LB-Explorer", to search for heterotic line bundle standard models arising from compactifications on smooth Calabi-Yau (CY) threefolds. We focus on $E_8\times E_8$ heterotic string theory compactifications on CY with abelian line bundles to produce $\text{SU}(5)\times \text{S}(\text{U}(1)^5)$ symmetry, whose $\text{SU}(5)$ can be further broken to an MSSM-like gauge group using appropriate discrete Wilson lines. We test the LB-Explorer environment on complete intersection Calabi-Yau (CICY) manifolds, though the neural network architecture naturally generalizes to any CY admitting a simplicial Mori cone and a freely-acting discrete symmetry. The LB-Explorer efficiently learns constraints on the line bundle sums, guaranteeing the $E_8$ gauge embedding, anomaly cancellation, poly-stability (supersymmetry), chirality of the spectrum, and the absence of exotic matter. Valid configurations can be subsequently filtered by imposing the missing constraints, such as the equivariant structure of the line bundle sum and further requirements on the particle spectrum. In this direction, we introduce a hybrid architecture incorporating CP-SAT solvers that aims to impose some of the conditions exactly by perturbing solutions found by the LB-Explorer. The versatility and scalability of the LB-Explorer make it a powerful tool for navigating the string landscape with a large number of moduli. The code and tools necessary to reproduce our findings are available at https://github.com/alexmininno/LB-Explorer

Problem

• string landscape
• heterotic compactifications
• line bundle standard models
• Calabi-Yau manifolds
• MSSM

Innovation

• Transformer-based Reinforcement Learning
• Line Bundle Standard Models
• Heterotic String Compactification
• Calabi-Yau Manifolds
• CP-SAT Solver