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We found some Lagrangian submanifolds of the adjoint semisimple orbit in two cases. For the first, the compact case, also known as the Generalized flag manifolds, we prove that the real flags can be seen as infinitesimally tight Lagrangian…
We give various realizations of the adjoint orbits of a semi-simple Lie group and describe their symplectic geometry. We then use these realizations to identify a family of Lagrangean submanifolds of the orbits.
We construct a Landau-Ginzburg model with the same data and symmetries as a $Z_2\times Z_2$ orbifold that corresponds to a class of realistic free-fermion models. Within the class of interest, we show that this orbifolding connects between…
This is the second part of a two--part series investigating bifurcation phenomena in autonomous Lagrangian systems and geodesic flows on Finsler and Riemannian manifolds. Building upon the abstract bifurcation theorems established in…
We formulate a constructive theory of noncommutative Landau-Ginzburg models mirror to symplectic manifolds based on Lagrangian Floer theory. The construction comes with a natural functor from the Fukaya category to the category of matrix…
The basics of operadic variational formalism is presented which is necessary when modeling the operadic systems. A general gauge theoretic approach to the abstract operads, based on the physical measurements concepts, is justified and…
We develop a $G$-equivariant Lagrangian Floer theory and obtain a curved $A_\infty$ algebra, and in particular a $G$-equivariant disc potential. We construct a Morse model, which counts pearly trees in the Borel construction $L_G$. When…
Let X be a compact oriented Riemannian manifold and let $\phi:X\to S^1$ be a circle-valued Morse function. Under some mild assumptions on $\phi$, we prove a formula relating: (a) the number of closed orbits of the gradient flow of $\phi$ of…
Recently, mirror symmetry is derived as T-duality applied to gauge systems that flow to non-linear sigma models. We present some of its applications to study quantum geometry involving D-branes. In particular, we show that one can employ…
In this note we describe a logarithmic version of mirror Landau-Ginzburg model for a semi-projective toric manifold and show the ring of state space of the Landau-Ginzburg model is isomorphic to the $\C$-valued cohomology of the toric…
The dynamics of gradient and Hamiltonian flows with particular application to flows on adjoint orbits of a Lie group and the extension of this setting to flows on a loop group are discussed. Different types of gradient flows that arise from…
A new method for constructing flows between distinct Landau-Ginzburg theories at fixed central charge is presented. The essential ingredient of the construction is an enlarged moduli space obtained by adding theories with zero central…
In this article we define intersection Floer homology for exact Lagrangian cobordisms between Legendrian submanifolds in the contactisation of a Liouville manifold, provided that the Chekanov-Eliashberg algebras of the negative ends of the…
Ginzburg-Landau (GL) equations and GL free energy for flux phase and superconductivity are derived microscopically from the $t-J$ model on a square lattice. Order parameter (OP) for the flux phase has direct coupling to a magnetic field, in…
We present the new explicit geometrical knowledge of the Landau-Ginzburg orbifolds, when a typical type of superpotential is considered. Relying on toric geometry, we show the one-to-one correspondence between some of the $(a,c)$ states…
Kontsevich suggested that the Landau-Ginzburg model presents a good formalism for homological mirror symmetry. In this paper we propose to investigate the LG theory from the viewpoint of Koopman-von Neumann's construction. New advances are…
Landau-Ginzburg mirror symmetry studies isomorphisms between A- and B-models, which are graded Frobenius algebras that are constructed using a weighted homogeneous polynomial $W$ and a related group of symmetries $G$ of $W$. It is known…
It is known that some equations of differential geometry are derived from variational principle in form of Euler-Lagrange equations. The equations of geodesic flow in Riemannian geometry is an example. Conversely, having Lagrangian…
We introduce a duality construction for toric Landau-Ginzburg models, applicable to complete intersections in toric varieties via the sigma model / Landau-Ginzburg model correspondence. This construction is shown to reconstruct those of…
Many fundamental structures of Riemannian geometry have found discrete counterparts for graphs or combinatorial ones for simplicial complexes. These include those discussed in this survey, Hodge theory, Morse theory, the spectral theory of…