Cluster Algebras, Atomic Bases, and String Theory

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Oftentimes in mathematics, a formula can arise from multiple perspectives. For example, the quadratic formula from high school algebra is used to find the roots of a polynomial equation but also indicates geometrically where a line and a parabola intersect. In research on so-called cluster algebras, formulas arise via a complicated but explicit process known as cluster mutation. In previous research, the PI developed simpler models that give an alternative way to compute these formulas and avoid the longer recursive procedure. This research project compares these models to those arising in string theory and geometry, revealing new features of all of these topics in the process, and creating exciting collaborations among physicists, mathematicians, and students.This project investigates various applications of cluster algebras, defined by Fomin and Zelevinsky in 2001, to topics in other fields of mathematics and physics. Cluster algebras are certain commutative algebras defined by binomial exchange relations introduced to study Lusztig's dual canonical bases. Since then, applications to many different topics have been discovered, including category theory, quiver representations, Teichmüller theory, discrete integrable systems, polyhedral combinatorics, and statistical physics, just to name a few. This research project will further investigate the connections to hyperbolic geometry and string theory. In particular, the PI will investigate scattering diagrams and brane tilings and their links to cluster algebras and cluster variables.