The closed-open string map for S1–invariant Lagrangians

Abstract

Abstract. Given a monotone Lagrangian submanifold invariant under a loop of Hamiltonian diffeomorphisms, we compute a piece of the closed-open string map into the Hochschild cohomology of the Lagrangian which captures the homology class of the loop's orbit.Our applications include split-generation and non-formality results for real Lagrangians in projective spaces and other toric varieties; a particularly basic example is that the equatorial circle on the 2-sphere carries a non-formal Fukaya A∞ algebra in characteristic two.

Generating the Fukaya categories of Hamiltonian 𝐺-manifolds

“…4 In a similar vein, we can prove that the real part L of a toric Fano variety X split-generates the Fukaya category when char(k) = 2, provided L is orientable. This was proved in certain cases by Tonkonog [64], including RP 2n+1 ⊂ CP 2n+1 (his methods also enable him to study some nonorientable cases). The real part L is known to have nonzero Floer cohomology [33].…”

“…Remark 7.2.2 Such generation results have been established when the superpotential has non-degenerate critical points in characteristic zero [54] and for Morse or A 2 -singularities in characteristic = 2, 3 [64]. Note that in Corollary 7.2.1 there is no assumption on nondegeneracy of the critical points of the superpotential or on the characteristic of the ground field.…”

Discrete and continuous symmetries in monotone Floer theory

“…Remark 5.4.4. The closed-open map for Lagrangians invariant under a loop γ of Hamiltonian diffeomorphisms has been studied by Charette-Cornea [6] and more recently by Tonkonog [39]. If S(γ) denotes the Seidel element in QH * (X) defined by γ, they showed that after setting the variable T to 1, and with the trivial local system, CO 0 (S(γ)) is equal to ±1 L , where the sign depends on the choice of spin structure.…”

“…Taking the trivial local system and standard spin structure, and setting T to 1, which corresponds to replacing each t j by 1 L , we obtain e N (η) = (−1) N • 1 L . It remains to check that the sign from [6,39] is also (−1) N .…”

“…Let γ denote an orbit of γ on L. A choice of spin structure on L induces a homotopy class of trivialisation of γ * T L, and by [39,Theorem 1.7] the sign of CO 0 (S(γ)) is +1 if and only if this homotopy class agrees with that defined by transport by γ. In our case, the latter corresponds to the identification T γ(θ) L = (g θ ) * T γ(0) L = g θ su(N − 1) • γ(0) for all θ.…”