  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 $(S^1 \times T)$-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.
