62 research outputs found
Fukaya–Seidel categories of Hilbert schemes and parabolic category O
No full textWe realise Stroppel’s extended arc algebra [13, 51] in the Fukaya–Seidel category of a natural Lefschetz fibration on the generic fibre of the adjoint quotient map on a type A nilpotent slice with two Jordan blocks, and hence obtain a symplectic interpretation of certain parabolic two-block versions of Bernstein–Gel’fand–Gel’fand category O. As an application, we give a new geometric construction of the spectral sequence from annular to ordinary Khovanov homology. The heart of the paper is the development of a cylindrical model to compute Fukaya categories of (affine open subsets of) Hilbert schemes of quasi-projective surfaces, which may be of independent interest
Tropically constructed Lagrangians in mirror quintic threefolds
No full textWe use tropical curves and toric degeneration techniques to construct closed embedded Lagrangian rational homology spheres in a lot of Calabi-Yau threefolds. The homology spheres are mirror dual to the holomorphic curves contributing to the Gromov-Witten (GW) invariants. In view of Joyce’s conjecture, these Lagrangians are expected to have special Lagrangian representatives and hence solve a special Lagrangian enumerative problem in Calabi-Yau threefolds.We apply this construction to the tropical curves obtained from the 2,875 lines on the quintic Calabi-Yau threefold. Each admissible tropical curve gives a Lagrangian rational homology sphere in the corresponding mirror quintic threefold and the Joyce’s weight of each of these Lagrangians equals the multiplicity of the corresponding tropical curve.As applications, we show that disjoint curves give pairwise homologous but non-Hamiltonian isotopic Lagrangians and we check in an example that >300 mutually disjoint curves (and hence Lagrangians) arise. Dehn twists along these Lagrangians generate an abelian subgroup of the symplectic mapping class group with that rank
Dehn twists and Lagrangian spherical manifolds
No full textWe study Dehn twists along Lagrangian submanifolds that are finite free quotients of spheres. We describe the induced auto-equivalences to the derived Fukaya category and explain their relations to mirror symmetry
Non-displaceable Lagrangian links in four-manifolds
No full textLet ω denote an area form on S2. Consider the closed symplectic 4-manifold M=(S2×S2,Aω⊕aω) with 0<a<A. We show that there are families of displaceable Lagrangian tori L0,x,L1,x⊂M, for x∈[0,1], such that the two-component link L0,x∪L1,x is non-displaceable for each x
Affine nil-Hecke algebras and quantum cohomology
No full textLet G be a compact, connected Lie group and T⊂G a maximal torus. Let (M, ω) be a monotone closed symplectic manifold equipped with a Hamiltonian action of G. We construct a module action of the affine nil-Hecke algebra H^S1×T∗(LG/T) on the S1×T-equivariant quantum cohomology of M, QH∗S1×T(M). Our construction generalizes the theory of shift operators for Hamiltonian torus actions [46,40]. We show that, as in the abelian case, this action behaves well with respect to the quantum connection. As an application of our construction, we show that the G-equivariant quantum cohomology QH∗G(M)defines a canonical holomorphic Lagrangian subvariety LG(M) to BFM(G∨C) in the BFM-space of the Langlands dual group, confirming an expectation of Teleman from [51]
A characterization of heaviness in terms of relative symplectic cohomology
No full textFor a compact subset of a closed symplectic manifold , we
prove that is heavy if and only if its relative symplectic cohomology over
the Novikov field is non-zero. As an application we show that if two compact
sets are not heavy and Poisson commuting, then their union is also not heavy. A
discussion on superheaviness together with some partial results are also
included.Comment: Added Corollary 1.10. To appear in the Journal of Topolog
Circular spherical divisor and their contact topology
No full textThis paper investigates the symplectic and contact topology associated to circular spherical divisors. We classify, up to toric equivalence, all concave circular spherical divisors D that can be embedded symplectically into a closed symplectic 4-manifold and show they are all realized as symplectic log Calabi-Yau pairs if their complements are minimal. We then determine the Stein fillability and rational homology type of all minimal symplectic fillings for the boundary torus bundles of such D. When D is anticanonical and convex, we give explicit betti number bounds for Stein fillings of its boundary contact torus bundle
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Fukaya–Seidel categories of Hilbert schemes and parabolic category
No full textCheuk Yu Mak, Ivan Smit
Quantitative Heegaard Floer cohomology and the Calabi invariant
No full textWe define a new family of spectral invariants associated to certain Lagrangian links in compact and connected surfaces of any genus. We show that our invariants recover the Calabi invariant of Hamiltonians in their limit. As applications, we resolve several open questions from topological surface dynamics and continuous symplectic topology: We show that the group of Hamiltonian homeomorphisms of any compact surface with (possibly empty) boundary is not simple; we extend the Calabi homomorphism to the group of hameomorphisms constructed by Oh and Müller, and we construct an infinite-dimensional family of quasi-morphisms on the group of area and orientation preserving homeomorphisms of the two-sphere.Our invariants are inspired by recent work of Polterovich and Shelukhin defining and applying spectral invariants, via orbifold Floer homology, for links composed of parallel circles in the two-sphere. A particular feature of our work is that it avoids the orbifold setting and relies instead on ‘classical’ Floer homology. This not only substantially simplifies the technical background but seems essential for some aspects (such as the application to constructing quasi-morphisms)
CUHK electronic theses & dissertations collection
No full textYu, Cheuk Yin Jandy.Thesis Ph.D. Chinese University of Hong Kong 2014.Includes bibliographical references (leaves 164-194).Abstracts also in Chinese.Title from PDF title page (viewed on 14, November, 2016)
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