Lagrangian Torus Fibers Research Articles

Chekanov torus and Gelfand–Zeitlin torus in S2 × S2

The Chekanov torus is the first known exotic torus, a monotone Lagrangian torus that is not Hamiltonian isotopic to the standard monotone Lagrangian torus. We explore the relationship between the Chekanov torus in S2×S2 and a monotone Lagrangian torus that had been constructed before Chekanov's construction [6]. We prove that the monotone Lagrangian torus fiber in a certain Gelfand–Zeitlin system is related to the Chekanov torus in S2×S2 by a symplectomorphism.

Non‐commutative amoebas

The group of isometries of the hyperbolic space H 3 $ \mathbb {H}^3$ is the 3-dimensional group PSL 2 ( C ) $\operatorname{PSL}_2(\mathbb {C})$ , which is one of the simplest non-commutative complex Lie groups. Its quotient by the subgroup SO ( 3 ) ⊂ PSL 2 ( C ) $\operatorname{SO}(3)\subset \operatorname{PSL}_2(\mathbb {C})$ naturally maps it back to H 3 $ \mathbb {H}^3$ . Each fiber of this map is diffeomorphic to the real projective 3-space R P 3 ${\mathbb {R}}{\mathbb {P}}^3$ . The resulting map PSL 2 ( C ) → H 3 $\operatorname{PSL}_2(\mathbb {C})\rightarrow \mathbb {H}^3$ can be viewed as the simplest non-commutative counterpart of the map Log : ( C × ) n → R n $\operatorname{Log}:(\mathbb {C}^\times )^n\rightarrow \mathbb {R}^n$ from the commutative complex Lie group ( C × ) n $(\mathbb {C}^\times )^n$ with the Lagrangian torus fibers that can be considered as a Liouville–Arnold type integrable system. Gelfand, Kapranov and Zelevinsky have introduced amoebas of algebraic varieties V ⊂ ( C × ) n $V\subset (\mathbb {C}^\times )^n$ as images Log ( V ) ⊂ R n $\operatorname{Log}(V)\subset \mathbb {R}^n$ . We define the amoeba of an algebraic subvariety of PSL 2 ( C ) $\operatorname{PSL}_2(\mathbb {C})$ as its image in H 3 $ \mathbb {H}^3$ . The paper surveys basic properties of the resulting hyperbolic amoebas and compares them against the commutative amoebas R n $\mathbb {R}^n$ .

Fukaya category of Grassmannians: Rectangles

We show that the monotone Lagrangian torus fiber of the Gelfand-Cetlin integrable system on the complex Grassmannian Gr(k,n) supports generators for all maximum modulus summands in the spectral decomposition of the Fukaya category over C, generalizing the example of the Clifford torus in projective space. We introduce an action of the dihedral group Dn on the Landau-Ginzburg mirror proposed by Marsh-Rietsch [27] that makes it equivariant and use it to show that, given a lower modulus, the torus supports nonzero objects in none or many summands of the Fukaya category with that modulus. The alternative is controlled by the vanishing of rectangular Schur polynomials at the n-th roots of unity, and for n=p prime this suffices to give a complete set of generators and prove homological mirror symmetry for Gr(k,p).

Potential functions on Grassmannians of planes and cluster transformations

With a triangulation of a planar polygon with $n$ sides, one can associate an integrable system on the Grassmannian of 2-planes in an $n$-space. In this paper, we show that the potential functions of Lagrangian torus fibers of the integrable systems associated with different triangulations glue together by cluster transformations. We also prove that the cluster transformations coincide with the wall-crossing formula in Lagrangian intersection Floer theory.

On an exotic Lagrangian torus in

We find a non-displaceable Lagrangian torus fiber in a semi-toric system which is superheavy with respect to a certain symplectic quasi-state. The proof employs both 4-dimensional techniques and those from symplectic field theory. In particular, our result implies Lagrangian$\mathbb{R}P^{2}$is not a stem in$\mathbb{C}P^{2}$, answering a question of Entov and Polterovich.

Holomorphic orbi-discs and Lagrangian Floer cohomology of symplectic toric orbifolds

We develop Floer theory of Lagrangian torus fibers in compact symplectic toric orbifolds. We first classify holomorphic orbi-discs with boundary on Lagrangian torus fibers. We show that there exists a class of basic discs such that we have one-to-one correspondences between a) smooth basic discs and facets of the moment polytope, and b) between basic orbi-discs and twisted sectors of the toric orbifold. We show that there is a smooth Lagrangian Floer theory of these torus fibers, which has a bulk deformation by fundamental classes of twisted sectors of the toric orbifold. We show by several examples that such bulk deformation can be used to illustrate the very rigid Hamiltonian geometry of orbifolds. We define its potential and bulk-deformed potential, and develop the notion of leading order potential.We study leading term equations analogous to the case of toric manifolds by Fukaya, Oh, Ohta, and Ono.

Toric degenerations of integrable systems on Grassmannians and polygon spaces

AbstractWe introduce a completely integrable system on the Grassmannian of 2-planes in ann-space associated with any triangulation of a polygon withnsides, and we compute the potential function for its Lagrangian torus fiber. The moment polytopes of this system for different triangulations are related by an integral piecewise-linear transformation, and the corresponding potential functions are related by its geometric lift in the sense of Berenstein and Zelevinsky.

Toric degenerations of integrable systems on Grassmannians and polygon spaces

AbstractWe introduce a completely integrable system on the Grassmannian of 2-planes in ann-space associated with any triangulation of a polygon withnsides, and we compute the potential function for its Lagrangian torus fiber. The moment polytopes of this system for different triangulations are related by an integral piecewise-linear transformation, and the corresponding potential functions are related by its geometric lift in the sense of Berenstein and Zelevinsky.

Potential functions via toric degenerations

We construct an integrable system on an open subset of a Fano manifold equipped with a toric degeneration, and compute the potential function for its Lagrangian torus fiber if the central fiber is a toric Fano variety admitting a small resolution.

The Ooguri-Vafa metric, holomorphic discs and wall-crossing

Recently, Gaiotto, Moore and Neitzke \cite{GMN08} proposed a new construction of hyperk\"{a}hler metrics. In particular, they gave a new construction of the Ooguri-Vafa metric, in which they came across certain formulas. We interpret those formulas as wall-crossing formulas that appear in the SYZ construction of instanton-corrected mirror manifolds. This reveals the close relation between the Ooguri-Vafa metric and nontrivial holomorphic discs with boundary in special Lagrangian torus fibers.

Toric degenerations of Gelfand–Cetlin systems and potential functions

We define a toric degeneration of an integrable system on a projective manifold, and prove the existence of a toric degeneration of the Gelfand–Cetlin system on the flag manifold of type A. As an application, we calculate the potential function for a Lagrangian torus fiber of the Gelfand–Cetlin system.

Non-displaceable Lagrangian submanifolds and Floer cohomology with non-unitary line bundle

We show that in many examples the non-displaceability of Lagrangian submanifolds by Hamiltonian isotopy can be proved via Lagrangian Floer cohomology with non-unitary line bundle. The examples include all monotone Lagrangian torus fibers in a toric Fano manifold (which was also proven by Entov and Polterovich via the theory of symplectic quasi-states) and some non-monotone Lagrangian torus fibers. We also extend the results by Oh and the author about the computations of Floer cohomology of Lagrangian torus fibers to the case of all toric Fano manifolds, removing the convexity assumption in the previous work.

Floer cohomology and disc instantons of Lagrangian torus fibers in Fano toric manifolds

In this paper, we first provide an explicit description of {\it all} holomorphic discs (``disc instantons'') attached to Lagrangian torus fibers of arbitrary compact toric manifolds, and prove their Fredholm regularity. Using this, we compute Fukaya-Oh-Ohta-Ono's (FOOO's) obstruction (co)chains and the Floer cohomology of Lagrangian torus fibers of Fano toric manifolds. In particular specializing to the formal parameter $T^{2\pi} = e^{-1}$, our computation verifies the folklore that FOOO's obstruction (co)chains correspond to the Landau-Ginzburg superpotentials under the mirror symmetry correspondence, and also proves the prediction made by K. Hori about the Floer cohomology of Lagrangian torus fibers of Fano toric manifolds. The latter states that the Floer cohomology (for the parameter value $T^{2\pi} = e^{-1}$) of all the fibers vanish except at a finite number, the Euler characteristic of the toric manifold, of base points in the momentum polytope that are critical points of the superpotential of the Landau-Ginzburg mirror to the toric manifold. In the latter cases, we also prove that the Floer cohomology of the corresponding fiber is isomorphic to its singular cohomology. We also introduce a restricted version of the Floer cohomology of Lagrangian submanifolds, which is a priori more flexible to define in general, and which we call the {\it adapted Floer cohomology}. We then prove that the adapted Floer cohomology of any non-singular torus fiber of Fano toric manifolds is well-defined, invariant under the Hamiltonian isotopy and isomorphic to the Bott-Morse Floer cohomology of the fiber.

Products of Floer Cohomology of Torus Fibers in Toric Fano Manifolds

We compute the ring structure of Floer cohomology groups of Lagrangian torus fibers in some toric Fano manifolds continuing the study of [CO]. Related A∞-formulas hold for a transversal choice of chains. Two different computations are provided: a direct calculation using the classification of holomorphic discs by Oh and the author in [CO], and another method by using an analogue of divisor equation in Gromov-Witten invariants to the case of discs. Floer cohomology rings are shown to be isomorphic to Clifford algebras, whose quadratic forms are given by the Hessians of functions W, which turn out to be the superpotentials of Landau-Ginzburg mirrors. In the case of , this proves the prediction made by Hori, Kapustin and Li by B-model calculations via physical arguments. The latter method also provides correspondence between higher derivatives of the superpotential of LG mirror with the higher products of the A∞(or L∞)-algebra of the Lagrangian submanifold.