Combinatorial mutations and deformations of dimer models

Abstract: The combinatorial mutation of a polytope was introduced in the context of the mirror symmetry of Fano manifolds for achieving the classification problem. This operation makes a given polytope another one while keeping some properties. In my talk, I will consider the combinatorial mutation of a polygon associated to a dimer model. A dimer model is a bipartite graph on the real two-torus, and the combinatorics of a dimer model gives rise to a certain lattice polygon. Also, a dimer model enjoys rich information regarding toric geometry associated to that polygon. It is known that for any lattice polygon P there is a dimer model whose associated polygon coincides with P. Thus, there also exists a dimer model giving the lattice polygon obtained as the combinatorial mutation of P. I will observe the relationship between a dimer model giving a lattice polygon P and the one giving the combinatorial mutation of P. In particular, I introduce the operation which I call the deformation of a dimer model, and show that this operation induces the combinatorial mutation of a polygon associated to a dimer model. This talk is based on a joint work with A. Higashitani.

algebraic geometrycombinatorics

Audience: researchers in the topic

Online Nottingham algebraic geometry seminar

Series comments: Online geometry seminar, typically held on Thursday. This seminar takes place online via Microsoft Teams on the Nottingham University "Algebraic Geometry" team.

For recordings of past talks, copies of the speaker's slides, or to be added to the Team, please visit the seminar homepage at: kasprzyk.work/seminars/ag.html