Cluster Algebras: Network Science and Machine Learning

Abstract

Cluster algebras have recently become an important player in mathematics and physics. In this work, we investigate them through the lens of modern data science, specifically with techniques from network science and machine-learning. Network analysis methods are applied to the exchange graphs for cluster algebras of varying mutation types. The analysis indicates that when the graphs are represented without identifying by permutation equivalence between clusters an elegant symmetry emerges in the quiver exchange graph embedding. The ratio between number of seeds and number of quivers associated to this symmetry is computed for finite Dynkin type algebras up to rank 5, and conjectured for higher ranks. Simple machine learning techniques successfully learn to differentiate cluster algebras from their seeds. The learning performance exceeds 0.9 accuracies between algebras of the same mutation type and between types, as well as relative to artificially generated data.

Machine Learning Algebraic Geometry for Physics

“…It was shown that the cluster algebra mutation rules can be realised as Seiberg dualities of quiver gauge theories, where Seiberg duality is a generalisation of the classical electro-magnetic duality for 4d N = 1 supersymmetric gauge theories, this interpretation is the focus of the work in [33] of which we summarise here. This work is also extended from quiver mutation to cluster mutation in [78].…”