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 \cite{CO}. Related \AI-formulas hold for transversal choice of chains. Two different computations are provided: a direct calculation using the classification of holomorphic discs by Oh and the author in \cite{CO}, and another method by using an {\it analogue of divisor equation} in Gromov-Witten invariants to thecase 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 \CP^n and \CP^1 \times \CP^1, 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 \AI(or \LI)-algebra of the Lagrangian submanifold.Comment: 26 pages, 3 fig; Statements about abstract perturbation and invariance withdrawn. Additional assumption on results about ring structur

Lagrangian fibers of Gelfand-Cetlin systems

Motivated by the study of Nishinou-Nohara-Ueda on the Floer thoery of Gelfand-Cetlin systems over complex partial flag manifolds, we provide a complete description of the topology of Gelfand-Cetlin fibers. We prove that all fibers are \emph{smooth} isotropic submanifolds and give a complete description of the fiber to be Lagrangian in terms of combinatorics of Gelfand-Cetlin polytope. Then we study (non-)displaceability of Lagrangian fibers. After a few combinatorial and numercal tests for the displaceability, using the bulk-deformation of Floer cohomology by Schubert cycles, we prove that every full flag manifold $\mathcal{F}(n)$ ($n \geq 3$) with a monotone Kirillov-Kostant-Souriau symplectic form carries a continuum of non-displaceable Lagrangian tori which degenerates to a non-torus fiber in the Hausdorff limit. In particular, the Lagrangian $S^3$-fiber in $\mathcal{F}(3)$ is non-displaceable the question of which was raised by Nohara-Ueda who computed its Floer cohomology to be vanishing.Comment: 84pages, lots of figure

Bogomol'nyi Solitons and Hermitian Symmetric Spaces

We apply the coadjoint orbit method to construct relativistic nonlinear sigma models (NLSM) on the target space of coadjoint orbits coupled with the Chern-Simons (CS) gauge field and study self-dual solitons. When the target space is given by Hermitian symmetric space (HSS), we find that the system admits self-dual solitons whose energy is Bogomol'nyi bounded from below by a topological charge. The Bogomol'nyi potential on the Hermitian symmetric space is obtained in the case when the maximal torus subgroup is gauged, and the self-dual equation in the $CP(N-1)$ case is explored. We also discuss the self-dual solitons in the non-compact $SU(1,1)$ case and present a detailed analysis for the rotationally symmetric solutions.Comment: 10 pages, 2 ps figures, Latex, A revised version to be published in Reports on Mathematical Physic