Quivers, Floer cohomology, and braid group actions

By Mikhail Khovanov, P. Seidel. 2000

322 citations recorded by the source corpus.

We consider the derived categories of modules over a certain family A_m of graded rings, and Floer cohomology of Lagrangian intersections in the symplectic manifolds which are the Milnor fibres of simple singularities of type A_m. We show that each of these two rather different objects encodes the topology of curves on an (m+1)-punctured disc. We prove that the braid group B_{m+1} acts faithfully on the derived category of A_m-modules and that it injects into the symplectic mapping class group of Milnor fibers. The philosophy behind our results is as follows. Using Floer cohomology, one should be able to associate to the Milnor fibre a triangulated category (its construction has not been carried out in detail). This triangulated category should contain a full subcategory which is equivalent, up to a slight difference in the grading, to the derived category of A_m-modules. The full embedding would connect the two occurrences of the braid group, thus explaining the similarity between them.

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/2dc9d9b454ba129a097936152281d04be8d66f83

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Citations in the corpus

A long exact sequence for symplectic Floer cohomology (2001, unknown): “An isomorphism of exact symplectic manifolds is a diffeomorphism φ : M → M ′ which is symplectic, satisfies φ∗θ′ = θ in some neighbourhood of ∂M , and such that [φ∗θ′ − θ] ∈ H1(M,∂M ;R) is zero.”
An open string analogue of Viterbo functoriality (2007, methodology): “We would like to thank Denis Auroux, Ivan Smith, and Katrin Wehrheim for valuable discussions.”
On partially wrapped Fukaya categories (2016, background): “This is straightforward in the stable case outside the central fiber, and in the unstable case one can argue as in [19].”
Symplectic Cohomology and q-Intersection Numbers (2010, background): “7) has been studied repeatedly, see for instance [21].”