Generating the Fukaya categories of Hamiltonian 𝐺-manifolds

Abstract

Let G be a compact Lie group and k be a field of characteristic p ≥ 0 such that H * (G) has no p-torsion if p > 0. We show that a free Lagrangian orbit of a Hamiltonian G-action on a compact, monotone, symplectic manifold X split-generates an idempotent summand of the monotone Fukaya category F (X; k) if and only if it represents a non-zero 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) through Koszul twisted complexes involving the zero-section and a cotangent fibre; and a functor D b W(T * G; k) → D b F (X − × X; k) canonically associated to the Hamiltonian G-action on X . We explore several examples which can be studied in a uniform manner including toric Fano varieties and certain Grassmannians.

The closed-open string map for S1–invariant Lagrangians

“…Consequently, these observations are a possible starting point for proving split-generation results for toric Fano varieties with other degenerate superpotentials. However, further development of this discussion seems both complicated and not particularly demanded, given the general results of [2,20].…”

A critical point analysis of Landau–Ginzburg potentials with bulk in Gelfand–Cetlin systems

“…The GC system admits nontorus Lagrangian GC fibers at the lower-dimensional strata of the GC polytope, which makes Floer theory of the system more interesting and challenging. Using non-Abelian symmetry or discrete symmetry, particular fibers of limited cases of Grassmannians have been investigated in [8], [9], and [28].…”

A simplicial construction of G‐equivariant Floer homology

“…These and later Floer‐type invariants are often enriched by various kinds of symmetries. For example, Seiberg–Witten Floer homology and cylindrical contact homology are intrinsically ‐equivariant theories [4, 23, 27], and Fukaya categories often carry actions of mapping class groups (for example, [22, 37]) and Lie algebras (for example, [8, 35]).…”