Generating the Fukaya categories of Hamiltonian 𝐺-manifolds

Abstract

Let G G be a compact Lie group, and let k k be a field of characteristic p ≥ 0 p \geq 0 such that H ∗ ( G ) H^*(G) has no p p -torsion if p > 0 p>0 . We show that a free Lagrangian orbit of a Hamiltonian G G -action on a compact, monotone, symplectic manifold X X split-generates an idempotent summand of the monotone Fukaya category F ( X ; k ) \mathcal {F}(X; k) if and only if it represents a nonzero object of that summand (slightly more general results are also provided). Our result is based on an explicit understanding of the wrapped Fukaya category W ( T ∗ G ; k ) \mathcal {W}(T^*G; k) through Koszul twisted complexes involving the zero-section and a cotangent fibre and on a functor D b W ( T ∗ G ; k ) → D b F ( X − × X ; k ) D^b \mathcal {W}(T^*G; k) \to D^b\mathcal {F}(X^{-} \times X; k) canonically associated to the Hamiltonian G G -action on X X . We explore several examples which can be studied in a uniform manner, including toric Fano varieties and certain Grassmannians.