Quantum multiplication operators for Lagrangian and orthogonal Grassmannians

Abstract

Abstract. In this article, we make a close analysis on quantum multiplication operators on the quantum cohomology rings of Lagrangian and orthogonal Grassmannians, and give an explicit description on all simultaneous eigenvectors and the corresponding eigenvalues for these operators. As a result, we show that Conjecture O of Galkin, Golyshev and Iritani holds for these manifolds.

Galkin's Lower bound Conjecure for Lagrangian and orthogonal Grassmannians

“…Then by Proposition 1.1, the δ 0 is an eigenvalue of [c 1 (M )] for LG(n) and OG(n). A direct proof of this also was given in [1].…”

“…Once a manifold M has property O, then δ 0 would become an eigenvalue of [c 1 (M )]. In fact, Galkin, Golyshev and Iritani conjectured that Fano manifolds have property O ( [7,9]), and it turned out that this is the case for many Fano manifolds [2,1,14,11]. In particular, we have Proposition 1.1 ( [2]).…”

Conjecture O holds for the odd symplectic Grassmannian

“…It is the main hypothesis needed for the statement of Gamma Conjectures I and II, which in turn are related to mirror symmetry on and generalize Dubrovin conjectures; we refer to for details. Property was proved for several Grassmannians of classical types and a complete proof was recently given for any homogeneous space . Other known cases include complete intersections in projective spaces , del Pezzo surfaces , few horospherical varieties , and a Bott–Samelson threefold .…”

Counting maximal Lagrangian subbundles over an algebraic curve

“…Then the integral coincides with a genus g Gromov-Witten invariant of the Lagrangian Grassmannian LG(C 2n ). Using results from [5], [6] and [7], the latter can be connected to a genus zero Gromov-Witten invariant of LG(C 2n ), whose closed formula is given by a Vafa-Intriligator-type formula.…”

“…where and [σ] denotes the quantum multiplication operator on qH * (LG(n), C) q=1 determined by σ. Then the formula follows from [7,Theorem 6.6] where the eigenvalues of [σ] were computed for an arbitrary σ ∈ qH * (LG(n), C) q=1 .…”