Shift operators and connections on equivariant symplectic cohomology

Abstract

We construct shift operators on equivariant symplectic cohomology which generalise the shift operators on equivariant quantum cohomology in algebraic geometry. That is, given a Hamiltonian action of the torus T , we assign to a cocharacter of T an endomorphism of (S 1 × T )-equivariant Floer cohomology based on the equivariant Floer Seidel map. We prove the shift operator commutes with a connection. This connection is a multivariate version of Seidel's q-connection on S 1 -equivariant Floer cohomology and generalises the Dubrovin connection on equivariant quantum cohomology. We prove that the connection is flat, which was conjectured by Seidel. As an application, we compute these algebraic structures for toric manifolds.

Affine nil-Hecke algebras and Quantum cohomology

“…defined by ( 60) is a map of H * (B T )-modules (compare [LJ,§3.16]). We will let H * T (M ) (σ[w]) be a copy of H * T (M ) equipped with the twisted R-module structure g…”

“…We next examine the interplay between our shift operators and the quantum connection (compare [I,Cor. 3.15] or [LJ,Theorem 1.6]). For simplicity, we consider a one-parameter connection along the direction −c T 1 (M ) ∈ H 2 T (M ) (an equivariant version of the "the anti-canonical line" in the terminology of [GGI]).…”

“…Thus, we are again reduced to the case of a co-character S σ as in [LJ,Theorem 3.6]. As strictly speaking the setup for Theorem 3.6 of loc.…”

“…Let E(σ) be the Seidel space and let c T 1,vert ∈ H 2 S 1 ×T (E(σ)) be the T -equvariant first Chern class of the vertical tangent bundle. Then the intertwining relation of [LJ,Theorem 3.4] Proof. (Sketch) These results are all proven in [BFM], but because they are elementary we indicate how they are proven.…”

Affine nil-Hecke algebras and quantum cohomology

“…The starting point for our work is Seidel's observation that for closed symplectic manifolds, a loop γ t of Hamiltonian symplectomorphisms induces an action on Hamiltonian Floer theory. Seidel's construction has been extended to include (convex) S 1 -actions on convex symplectic manifolds in [R,LJ2]. The new feature, compared to the closed case, is that a Hamiltonian loop can modify the behavior of the Hamiltonian at infinity.…”