Connections on equivariant Hamiltonian Floer cohomology

Seidel, P.

doi:10.4171/cmh/445

Cited by 6 publications

(10 citation statements)

References 30 publications

(56 reference statements)

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“…There is an operation on S 1 -equivariant symplectic cohomology, which we'll call the Gauss-Manin connection. It exists without any additional assumptions, and is canonical [45]. Even though it is therefore somewhat different from our ∇ c , one expects the two to be related via a suitable intermediate object.…”

Section: A Sketch Of the Equivariant Theorymentioning

confidence: 92%

Fukaya A ∞-structures Associated to Lefschetz Fibrations. II

Seidel

2017

Algebra, Geometry, and Physics in the 21st Century

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Floer cohomology groups are usually defined over a field of formal functions (a Novikov field). Under certain assumptions, one can equip them with connections, which means operations of differentiation with respect to the Novikov variable. This allows one to write differential equations for Floer cohomology classes. Here, we apply that idea to symplectic cohomology groups associated to Lefschetz fibrations, and establish a relation with enumerative geometry.

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Section: A Sketch Of the Equivariant Theorymentioning

confidence: 92%

Fukaya A ∞-structures Associated to Lefschetz Fibrations. II

Seidel

2017

Algebra, Geometry, and Physics in the 21st Century

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“…Our definition is close to those of [BO17, Sections 2.2-2.3] and [Gut18, Section 2.3], except we use cohomological conventions and a Novikov ring. Our conventions are almost identical to Zhao's periodic symplectic cohomology in [Zha19, Zha16] and Seidel's definition in [Sei18], except that we use a direct sum convention for cohomology rather than a direct product (and we use a Novikov ring). Importantly, we do not use Z[u, u −1 ]/(uZ[u]) in our coefficient ring unlike [Sei08, Section 8b] and [MR18, Appendix B]; indeed this is not possible with our module operation.…”

Section: Equivariant Floer Theorymentioning

confidence: 99%

An intertwining relation for equivariant Seidel maps

Liebenschutz-Jones¹

2020

Preprint

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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.

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“…Inspired by their construction (but different from theirs as noted in Remark 1.1), we extend a GGM connection to a "connection in the zdirection". In different contexts, similar constructions have been considered by many people (e.g., [22, §4.1], [32] and see also Remark 1.2). In §5.4, we continue to use the same notations as in §5.3.…”

Section: Maurer-cartan Elementsmentioning

confidence: 91%

“…There are related works, e.g. in [18], [32] motivated also by symplectic geometry. In [18, Section 4.2] Ganatra-Perutz-Sheridan claims the compatibility of the Getzler-Gauss-Manin connection and Dubrovin's quantum connection under the cyclic open-closed map.…”

Section: Introductionmentioning

confidence: 99%

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Meromorphic connections in filtered $A_{\infty}$ categories

Ohta¹,

Sanda²

2019

Preprint

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In this note, introducing notions of CH module, CH morphism and CH connection, we define a meromorphic connection in the "z-direction" on periodic cyclic homology of an A∞ category as a connection on cohomology of a CH module. Moreover, we study and clarify compatibility of our meromorphic connections under a CH module morphism preserving CH connections at chain level. Our motivation comes from symplectic geometry. The formulation given in this note designs to fit algebraic properties of open-closed maps in symplectic geometry.

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Connections on equivariant Hamiltonian Floer cohomology

Abstract: We construct connections on S 1 -equivariant Hamiltonian Floer cohomology, which differentiate with respect to certain formal parameters.

Cited by 6 publications

References 30 publications

Fukaya A ∞-structures Associated to Lefschetz Fibrations. II

Fukaya A ∞-structures Associated to Lefschetz Fibrations. II

An intertwining relation for equivariant Seidel maps

Meromorphic connections in filtered $A_{\infty}$ categories

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