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Workshop: Symmetry and Geometry in Neural Representations

Haldane Bundles: A Dataset for Learning to Predict the Chern Number of Line Bundles on the Torus

Cody Tipton · Elizabeth Coda · Davis Brown · Alyson Bittner · Caitlin Hutten · Grayson Jorgenson · Tegan Emerson · Henry Kvinge

Abstract: Characteristic classes, which are abstract topological invariants associated with vector bundles, have become an important notion in modern physics with surprising real-world consequences. As a representative example, the incredible properties of topological insulators, which are insulators in their bulk but conductors on their surface, can be completely characterized by a specific characteristic class associated with their electronic band structure, the first Chern class. Given their importance to next generation computing and the computational challenge of calculating them using first-principles approaches, there is a need to develop machine learning approaches to predict the characteristic classes associated with a material system. To aid in this program we introduce the *Haldane bundle dataset*, which consists of synthetically generated complex line bundles on the $2$-torus. We envision this dataset, which is not as challenging as noisy and sparsely measured real-world datasets but (as we show) still difficult for off-the-shelf architectures, to be a testing ground for architectures that incorporate the rich topological and geometric priors underlying characteristic classes.

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