Cluster algebras, 3D integrable systems and 3D gauge theories

Solutions of Zamolodchikov's tetrahedron equation define integrable 3D lattice models in statistical mechanics, just as solutions of the Yang-Baxter equation define integrable 2D lattice models. I will explain how we can construct solutions of the tetrahedron equation using quantum cluster algebras and how they are related to 3D supersymmetric gauge theories. This is based on joint work with Xiao-yue Sun, Rei Inoue, Atsuo Kuniba and Yuji Terashima.