TY - JOUR
TI - On the Fukaya category of Grassmannians
DO - https://doi.org/doi:10.7282/t3-veb9-f129
PY - 2021
AB - The Grassmannian of k-dimensional planes in a complex n-dimensional vector space has a natural symplectic structure, that one can use to construct a deformation of its cohomology ring. We begin the study of its Fukaya category, which is a refinement of the category of modules over this ring. When n is a prime number, we prove that a fiber of an integrable system introduced by Guillemin-Sternberg split-generates the Fukaya category. If n is not prime this generally fails, and we construct examples of Lagrangian tori that support nonzero objects in the missing part of the Fukaya category. The tori are parametrized by seeds of a cluster algebra in the sense of Fomin-Zelevinsky, and have associated Laurent polynomials with positive integer coefficients. We program a random walk that computes these Laurent polynomials explicitly, and observe that they encode the open genus zero Gromov-Witten invariants of the tori in some cases. We put forth a conjecture on the general meaning of the Laurent polynomials, which can be considered as a Symplectic Field Theory interpretation of the notion of scattering diagram proposed by Gross-Hacking-Keel-Kontsevich.
KW - Mathematics
KW - Cluster algebras
LA - English
ER -