Fukaya category of Grassmannians: rectangles

TL;DR

This paper demonstrates that certain Lagrangian tori in Grassmannians generate all spectral summands of the Fukaya category, establishing homological mirror symmetry for Grassmannians of prime dimension.

Contribution

It introduces an equivariant Landau-Ginzburg mirror and proves homological mirror symmetry for Grassmannians of prime dimension using Schur polynomial vanishing criteria.

Findings

Lagrangian torus fibers generate all spectral summands in Grassmannians.

The dihedral group action makes the mirror equivariant, aiding analysis.

Homological mirror symmetry is proven for Grassmannians of prime dimension.

Abstract

We show that the monotone Lagrangian torus fiber of the Gelfand-Cetlin integrable system on the complex Grassmannian $\operatorname{Gr}(k,n)$ supports generators for all maximum modulus summands in the spectral decomposition of the Fukaya category over $\mathbb{C}$, generalizing the example of the Clifford torus in projective space. We introduce an action of the dihedral group $D_n$ on the Landau-Ginzburg mirror proposed by Marsh-Rietsch that makes it equivariant and use it to show that, given a lower modulus, the torus supports nonzero objects in none or many summands of the Fukaya category with that modulus. The alternative is controlled by the vanishing of rectangular Schur polynomials at the $n$-th roots of unity, and for $n=p$ prime this suffices to give a complete set of generators and prove homological mirror symmetry for $\operatorname{Gr}(k,p)$.