Equivariant Seidel maps and a flat connection on equivariant symplectic cohomology

Thesis

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 a few specific examples and provide a method to compute them for toric manifolds using the moment polytope.

Authors: Liebenschutz-Jones, T

• DOI: 10.5287/ora-0zb8nkke0
• Type of award: DPhil
• Level of award: Doctoral
• Awarding institution: University of Oxford
• Language: English
• Keywords: equivariant symplectic cohomology; floer cohomology; Seidel map
• Subjects: Symplectic geometry; Geometry, Algebraic; Symplectic and contact topology