Cluster Algebras and Mirror Symmetry for Homogeneous Spaces

Abstract

Homogeneous spaces lie at the intersection of various fields of study, and as a result possess an incredibly rich structure. This thesis focuses in particular on the interactions between cluster algebras and mirror symmetry in the context of homogeneous spaces.

Rietsch’s construction of Landau-Ginzburg (LG) models for homogeneous spaces motivated studies of various mirror symmetry statements for homogeneous spaces, such as Grassmannians Gr(k,n) which are homogeneous spaces for the special linear group SL(n) . In nearly all of these studies, the authors preferred to work with an equivalent LG model presented in terms of coordinates rather than with Rietsch’s original Lie-theoretic formulation. However, coordinate formulations of Rietsch’s LG models were only available for certain special cases, and were difficult to generalize. In joint work with Peter Spacek, which forms the first part of this thesis, we give a coordinate presentation of Rietsch’s LG models for the Cayley plane and Freudenthal variety, which are homogeneous spaces for the exceptional Lie groups of types E6 and E7, respectively. Furthermore, we are currently working to extend our methods to the more general family of cominuscule homogeneous spaces, and present preliminary results in this direction in this thesis.

In some works mentioned above, cluster algebras were used to facilitate proofs or explain certain phenomena. These interesting connections between mirror symmetry and cluster algebras for homogeneous spaces are most well-studied for the Grassmannians, for example in the works of Marsh and Rietsch and of Rietsch and Williams. We expect this connection is not specific to Grassmannians, but rather a general feature of homogeneous spaces, and in this direction we present exploratory work for the Lagrangian and orthogonal Grassmannians, which are homogeneous spaces for the symplectic group Sp(2n) and the special orthogonal group SO(2n+1), respectively. We hope to greatly develop this connection further.

There have also been remarkable connections between cluster algebras and integrability in the context of Grassmannians Gr(k,n). The recent works of Kodama and Williams as well as of Abenda and Grinevich study the relationship between soliton solutions to the KP equation and the structure of the (totally nonnegative) Grassmannian. These connections remain somewhat mysterious, and in order to gain further insight into this relationship, we study the problems of identifying commuting differential operators and reconstructing solutions to the KP equation from water waves. Commuting differential operators, particularly in the context of of pseudo-differential operators and the Sato Grassmannian, have deep connections to the KP equation and algebraic curves. Furthermore, Krichever showed how to construct solutions to the KP equation using algebraic curves, and understanding the inverse problem will be helpful in studying the combinatorial structure of solutions to the KP equation.