Symplectic cohomology and q-intersection numbers

Abstract

Given a symplectic cohomology class of degree 1, we define the notion of an equivariant Lagrangian submanifold. The Floer cohomology of equivariant Lagrangian submanifolds has a natural endomorphism, which induces a grading by generalized eigenspaces. Taking Euler characteristics with respect to the induced grading yields a deformation of the intersection number. Dehn twists act naturally on equivariant Lagrangians. Cotangent bundles and Lefschetz fibrations give fully computable examples. A key step in computations is to impose the "dilation" condition stipulating that the BV operator applied to the symplectic cohomology class gives the identity. Equivariant Lagrangians mirror equivariant objects of the derived category of coherent sheaves.

32 pages, 9 figures, expanded introduction, added details of example 7.5, added discussion of signs

References in corpus (5)

• Functors and Computations in Floer homology with Applications Part II
• Mirror symmetry and T-duality in the complement of an anticanonical divisor
• Fukaya categories and deformations
• A geometric criterion for generating the Fukaya category
• Localization for involutions in Floer cohomology

Cited by in corpus (25)

• Symplectic cohomology and duality for the wrapped Fukaya category
• On the symplectic cohomology of log Calabi-Yau surfaces
• The symplectic arc algebra is formal
• Cyclic homology, $S^1$-equivariant Floer cohomology, and Calabi-Yau structures
• Lagrangian homology spheres in (A_m) Milnor fibres via C^*-equivariant A_infinity modules
• Picard-Lefschetz theory and dilating C^*-actions
• Disjoinable Lagrangian spheres and dilations
• A Log PSS morphism with applications to Lagrangian embeddings
• Symplectic fillings of asymptotically dynamically convex manifolds I
• Symplectic fillings of asymptotically dynamically convex manifolds II--$k$-dilations
• Homological mirror symmetry for Milnor fibers of simple singularities
• Floer cohomology of $\mathfrak{g}$-equivariant Lagrangian branes
• On the embedding complexity of Liouville manifolds
• Koszul duality via suspending Lefschetz fibrations
• Knot Categorification from Mirror Symmetry, Part II: Lagrangians
• Exact Calabi-Yau categories and odd-dimensional Lagrangian spheres
• Disjoinable Lagrangian tori and semisimple symplectic cohomology
• Exact orbifold fillings of contact manifolds
• Exotic families of symplectic manifolds with Milnor fibers of $ADE$-type
• On the minimal symplectic area of Lagrangians
• Bigrading the symplectic Khovanov cohomology
• Nonexistence of exact Lagrangian tori in affine conic bundles over ${\mathbb{C}}^n$
• A landscape of contact manifolds via rational SFT
• Persistence of unknottedness of clean Lagrangian intersections
• A symplectic prolegomenon