Newton-Okounkov bodies, cluster duality, and mirror symmetry for Grassmannians

Abstract

We use cluster structures and mirror symmetry to explicitly describe a natural class of Newton-Okounkov bodies for Grassmannians. We consider the Grassmannian $X=G{r}_{n-k}({\mathbb{C}}^n)$, as well as the mirror dual Landau-Ginzburg model $({\check{X}}^{\circ },W_q:{\check{X}}^{\circ }\to \mathbb{C})$, where ${\check{X}}^{\circ }$ is the complement of a particular anti-canonical divisor in a Langlands dual Grassmannian $\check{X}=G{r}_k(({\mathbb{C}}^n{)}^{\ast })$, and the superpotential W_q has a simple expression in terms of Plücker coordinates. Grassmannians simultaneously have the structure of an $\mathcal{A}$-cluster variety and an $\mathcal{X}$-cluster variety. Given a cluster seed G, we consider two associated coordinate systems: a $\mathcal{X}$-cluster chart ${\Phi }_G:({\mathbb{C}}^{\ast }{)}^{k(n-k)}\to X^{\circ }$ and a $\mathcal{A}$-cluster chart ${\Phi }_G^{\vee }:({\mathbb{C}}^{\ast }{)}^{k(n-k)}\to {\check{X}}^{\circ }$. To each $\mathcal{X}$-cluster chart ${\Phi }_G$ and ample `boundary divisor' $D$ in $X\setminus X^{\circ }$, we associate a Newton-Okounkov body ${\Delta }_G(D)$ in ${\mathbb{R}}^{k(n-k)}$, which is defined as the convex hull of rational points. On the other hand using the $\mathcal{A}$-cluster chart ${\Phi }_G^{\vee }$ on the mirror side, we obtain a set of rational polytopes, described by inequalities, by writing the superpotential $W_q$ in the $\mathcal{A}$-cluster coordinates, and then "tropicalising". Our main result is that the Newton-Okounkov bodies ${\Delta }_G(D)$ and the polytopes obtained by tropicalisation coincide. As an application, we construct degenerations of the Grassmannian to toric varieties corresponding to these Newton-Okounkov bodies. Additionally, when $G$ corresponds to a plabic graph, we give a formula for the lattice points of the Newton-Okounkov bodies, which has an interpretation in terms of quantum Schubert calculus.

55 pages, many figures; to appear in Duke

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