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 $\mathbb{X}=G{r}_{n-k}({\mathbb{C}}^n)$. 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 $\check{\mathbb{X}}=G{r}_k(({\mathbb{C}}^n{)}^{\ast })$, and the superpotential $W_q$ has a simple expression in terms of Plücker 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 ${\Phi }_G:({\mathbb{C}}^{\ast }{)}^{k(n-k)}\to \mathbb{X}$ in our $A$-model, and using work of Scott we have a cluster chart ${\Phi }_G^{\vee }:({\mathbb{C}}^{\ast }{)}^{k(n-k)}\to \check{\mathbb{X}}$ in our $B$-model. To each positive chart ${\Phi }_G$ and choice of positive integer $r$, we associate a polytope $N{O}_G^r$, 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 $N{O}_G^r$ and $Q_G^r$ coincide.

The paper has a gap in not considering the non-integral case and is subsumed by 1712.00447, where in particular this gap is filled

References in corpus (3)

• Total positivity, Grassmannians, and networks
• Matching polytopes, toric geometry, and the non-negative part of the Grassmannian
• Fock-Goncharov conjecture and polyhedral cones for $U\subset S{L}_n$ and base affine space $S{L}_n/U$

Cited by in corpus (7)

• The Berenstein-Zelevinsky quantum cluster algebra conjecture
• Calabi--Yau Operators
• Donaldson-Thomas Transformation of Double Bruhat Cells in General Linear Groups
• Potential functions on Grassmannians of planes and cluster transformations
• Toric degenerations of Gr(2,n) and Gr(3,6) via plabic graphs
• Tropicalization of Positive Grassmannians
• Tropical critical points of the superpotential of a flag variety