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.

The Seidel maps are two maps associated to a Hamiltonian circle action on a convex symplectic manifold, one on Floer cohomology and one on quantum cohomology. We extend their definitions to S 1 -equivariant Floer cohomology and S 1 -equivariant quantum cohomology based on a construction of Maulik and Okounkov. The S 1 -action used to construct S 1 -equivariant Floer cohomology changes after applying the equivariant Seidel map (a similar phenomenon occurs for S 1equivariant quantum cohomology). We show the equivariant Seidel map on S 1equivariant quantum cohomology does not commute with the S 1 -equivariant quantum product, unlike the standard Seidel map. We prove an intertwining relation which completely describes the failure of this commutativity as a weighted version of the equivariant Seidel map. We will explore how this intertwining relationship may be interpreted using connections in an upcoming paper. We compute the equivariant Seidel map for rotation actions on the complex plane and on complex projective space, and for the action which rotates the fibres of the tautological line bundle over projective space. Through these examples, we demonstrate how equivariant Seidel maps may be used to compute the S 1 -equivariant quantum product and S 1 -equivariant symplectic cohomology.