Cluster duality and mirror symmetry for Grassmannians

Abstract

In this article we use the cluster structure on the Grassmannian and the combinatorics of plabic graphs to exhibit a new aspect of mirror symmetry for Grassmannians in terms of polytopes. For our $A$-model, we consider the Grassmannian $\mathbb X=Gr_{n-k}(\mathbb{C}^n)$. The $B$-model is a Landau-Ginzburg model $(\check{\mathbb X}^\circ, W_q:\check{\mathbb X}^\circ \to \mathbb{C})$, where $\check{\mathbb X}^\circ$ is the complement of a particular anti-canonical divisor in a Langlands dual Grassmannian $\check{\mathbb X} = Gr_k((\mathbb{C}^n)^*)$, and the superpotential $W_q$ has a simple expression in terms of Pl\"ucker coordinates, see [MarshRietsch]. From a given plabic graph $G$ we obtain two coordinate systems: using work of Postnikov and Talaska we have a positive chart $\Phi_G:(\mathbb{C}^*)^{k(n-k)}\to \mathbb X$ in our $A$-model, and using work of Scott we have a cluster chart $\Phi_G^{\vee}:(\mathbb{C}^*)^{k(n-k)}\to \check{\mathbb X}$ in our $B$-model. To each positive chart $\Phi_G$ and choice of positive integer $r$, we associate a polytope $NO_G^r$, which we construct as the convex hull of a set of integer lattice points. This polytope is an example of a Newton-Okounkov polytope associated to the line bundle $\mathcal O(r)$ on $\mathbb X$. On the other hand, using the cluster chart $\Phi_G^{\vee}$ and the same positive integer $r$, we obtain a polytope $Q_G^r$ -- described in terms of inequalities -- by "tropicalizing" the composition $W_{t^r}\circ \Phi_G^{\vee}$. Our main result is that the polytopes $NO_G^r$ and $Q_G^r$ coincide.