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

TL;DR

This paper explicitly describes Newton-Okounkov bodies for Grassmannians using cluster structures and mirror symmetry, showing they coincide with tropicalized polytopes and enabling toric degenerations.

Contribution

It introduces a new explicit correspondence between Newton-Okounkov bodies and tropical polytopes via cluster charts for Grassmannians, linking mirror symmetry and quantum Schubert calculus.

Findings

Newton-Okounkov bodies coincide with tropicalized polytopes.

Constructs toric degenerations of Grassmannians.

Provides formulas for lattice points related to quantum Schubert calculus.

Abstract

We use cluster structures and mirror symmetry to explicitly describe a natural class of Newton-Okounkov bodies for Grassmannians. We consider the Grassmannian $X=Gr_{n-k}(\mathbb C^n)$, 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 $\check{X} = Gr_k((\mathbb C^n)^*)$, and the superpotential W_q has a simple expression in terms of Pl\"ucker 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 $\Phi_G:(\mathbb C^*)^{k(n-k)}\to X^{\circ}$ and a $\mathcal A$-cluster chart $\Phi_G^{\vee}:(\mathbb C^*)^{k(n-k)}\to \check{X}^\circ$. 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.