An intertwining relation for equivariant Seidel maps

Abstract

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.

Shift operators and connections on equivariant symplectic cohomology

“…Theorem 3.4 is the T -equivariant version of the S 1 -equivariant intertwining relation that we showed in [LJ20], and the proof is almost identical. The new proofs in Section 4 use similar ideas.…”

“…for any two cocharacters σ, σ ′ . We introduced the S 1 -equivariant Floer Seidel map F S S 1 (σ, µ) in our previous work [LJ20]. This map combines the identity map on BS 1 with the Floer Seidel map on M .…”

“…The algebrogeometric technique of virtual localisation does not work in equivariant Floer cohomology. In [LJ20], we gave a new Morse-theoretic proof of the intertwining relation which does not use virtual localisation. This proof gives the relation as the boundary of a 1-dimensional moduli space which is made up of a configuration on ES 1 = S ∞ and a configuration on M which depends on the configuration on ES 1 .…”

“…Notation. In [LJ20], we only looked at S 1 -equivariant constructions, denoting S 1 -equivariant invariants with an E (EQS, EQH * , etc.). In this paper, we consider S 1 -equivariant, T -equivariant and T -equivariant constructions, so we specify the group in this paper's notation (QS S 1 , QH * T , etc.).…”

“…It turns out that the point ε min and the entire second flowline are uniquely determined by ε and z 0 , so we discard this second flowline from the moduli space without losing any information. Using a homotopy, we decouple the flowline to c from the rest of the configuration, and the resulting moduli spaces yield the maps y 0 and WQS T (σ, µ, α) respectively (more details in [LJ20]).…”