Abstract: |
The tetrahedron equation, as the 3-dimensional analogue of the Yang-Baxter equation, plays a central role in 3-dimensional integrable lattice models and (2+1)-dimensional quantum field theories. In this talk, I will explain how to construct solutions to the tetrahedron equation from the cluster transformations in cluster algebra, based on the work with my supervisor Junya Yagi, as detailed in arXiv:2211.10702. Additionally, I will briefly explain how these solutions can be identified with the partition functions of some 3d N=2 gauge theories on the three-dimensional squashed sphere. I will also introduce the ongoing work with Rei Inoue, Atsuo Kuniba, Yuji Terashima, and Junya Yagi, in which we discovered that our approach based on cluster algebra can reproduce many important known solutions of the tetrahedron equation. |