158 research outputs found
Functoriality for Lagrangian correspondences in Floer theory
Using quilted Floer cohomology and relative quilt invariants, we define a
composition functor for categories of Lagrangian correspondences in monotone
and exact symplectic Floer theory. We show that this functor agrees with
geometric composition in the case that the composition is smooth and embedded.
As a consequence we obtain 'categorification commutes with composition' for
Lagrangian correspondences.Comment: minor corrections and updated references; original 120 page preprint
got split into 4 parts - this is one of the
A wall-crossing formula for Gromov-Witten invariants under variation of git quotient
We prove a quantum version of Kalkman's wall-crossing formula comparing
Gromov-Witten invariants on geometric invariant theory (git) quotients related
by a change in polarization. The wall-crossing terms are gauged Gromov-Witten
invariants with smaller structure group. As an application, we show that the
graph Gromov-Witten potentials of quotients related by wall-crossings of
crepant type are equivalent up to a distribution in the quantum parameter that
is almost everywhere zero. This is a version of the crepant transformation
conjecture of Li-Ruan, Bryan-Graber, Coates-Ruan etc. in cases where the
crepant transformation is obtained by variation of git.Comment: 64 pages, 1 figure. Expanded and clarified exposition in a number of
places in response to referee comment
Floer Cohomology and Geometric Composition of Lagrangian Correspondences
We prove an isomorphism of Floer cohomologies under geometric composition of
Lagrangian correspondences in exact and monotone settings.Comment: minor corrections, in particular more precise formulation of
monotonicity assumption
Fukaya categories of blowups
We compute the Fukaya category of the symplectic blowup of a compact rational
symplectic manifold at a point in the following sense: Suppose a collection of
Lagrangian branes satisfy Abouzaid's criterion for split-generation of a
bulk-deformed Fukaya category of cleanly-intersecting Lagrangian branes. We
show that for a small blow-up parameter, their inverse images in the blowup
together with a collection of branes near the exceptional locus split-generate
the Fukaya category of the blowup. This categorifies a result on quantum
cohomology by Bayer and is an example of a more general conjectural description
of the behavior of the Fukaya category under transitions occuring in the
minimal model program, namely that mmp transitions generate additional
summands.Comment: 82 page
Augmentation varieties and disk potentials
We elaborate on a suggestion of Aganagic-Ekholm-Ng-Vafa, that in order for
Lagrangian fillings such as the Harvey-Lawson filling to define augmentations
of Chekanov-Eliashberg differential graded algebras, one should count
configurations of holomorphic disks connected by gradient trajectories. We
propose a definition of the Chekanov-Eliashberg dga in higher dimensions which
includes as generators both Reeb chords and the space of chains on the
Legendrian, similar to the definition of immersed Lagrangian Floer theory whose
generators are chains on the Lagrangian as well as self-intersection points. We
prove that for connected Legendrian covers of monotone Lagrangian tori, the
augmentation variety in this model is equal to the image of the zero level set
of the disk potential, as suggested by Dimitroglou-Rizell-Golovko. In
particular, we show that Legendrian lifts of Vianna's exotic tori are not
Legendrian isotopic, as conjectured in Dimitroglou-Rizell-Golovko. Using
related ideas, we show that the Legendrian lift of the Clifford torus admits no
exact fillings, extending the results of Dimitroglou-Rizell and Treumann-Zaslow
in dimension two. We consider certain disconnected Legendrians, and show,
similar to another suggestion of Aganagic-Ekholm-Ng-Vafa, that the components
of the augmentation variety correspond to certain partitions and each component
is defined by a (not necessarily exact) Lagrangian filling. An adaptation of
the theory of holomorphic quilts shows that the cobordism maps associated to
bounding chains are independent of all choices up to chain homotopy.Comment: 157 page
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