Gelfand-Cetlin System Research Articles

The double Gelfand–Cetlin system, invariance of polarization, and the Peter–Weyl theorem

The bundle map T⁎U(n)⟶U(n) provides a real polarization of the cotangent bundle T⁎U(n), and yields the geometric quantization Q1(T⁎U(n))=L2(U(n)). We use the Gelfand–Cetlin systems of Guillemin and Sternberg to show that T⁎U(n) has a different real polarization with geometric quantization Q2(T⁎U(n))=⨁αVα⊗Vα⁎, where the sum is over all dominant integral weights α of U(n). The Peter–Weyl theorem, which states that these two quantizations are isomorphic, may therefore be interpreted as an instance of “invariance of polarization” in geometric quantization.

Abelianization and the Duistermaat–Heckman theorem

AbstractWe use the abelianization theorem of Crooks and Weitsman (2022) to prove a non‐abelian generalization of the Duistermaat–Heckman theorem for measures. Our main technical tools include the Gelfand–Cetlin data of Crooks and Weitsman (2022), examples of which are the Gelfand–Cetlin systems of Guillemin–Sternberg and generalizations thereof due to Hoffman–Lane.

A Leray–Serre spectral sequence for Lagrangian Floer theory

We consider symplectic fibrations as in Guillemin-Lerman-Sternberg, and derive a spectral sequence to compute the Floer cohomology of certain fibered Lagrangians sitting inside a compact symplectic fibration with small monotone fibers and a rational base. We show if the Floer cohomology with field coefficients of the fiber Lagrangian vanishes, then the Floer cohomology with field coefficients of the total Lagrangian also vanishes. We give an application to certain non-torus fibers of the Gelfand-Cetlin system in Flag manifolds, and show that their Floer cohomology vanishes.

A critical point analysis of Landau–Ginzburg potentials with bulk in Gelfand–Cetlin systems

Using the bulk deformation of Floer cohomology by Schubert classes and non-Archimedean analysis of Fukaya–Oh–Ohta–Ono’s bulk-deformed potential function, we prove that every complete flag manifold Fl(n) (n≥3) with a monotone Kirillov–Kostant–Souriau (KKS) symplectic form carries a continuum of nondisplaceable Lagrangian tori which degenerates to a nontorus fiber in the Hausdorff limit. In particular, the Lagrangian S3-fiber in Fl(3) is nondisplaceable, answering a question raised by Nohara and Ueda who computed its Floer cohomology to be vanishing.

Lagrangian fibers of Gelfand-Cetlin systems

A Gelfand-Cetlin system is a completely integrable system defined on a partial flag manifold whose image is a rational convex polytope called a Gelfand-Cetlin polytope. Motivated by the study of Nishinou-Nohara-Ueda [24] on the Floer theory of Gelfand-Cetlin systems, we provide a detailed description of topology of Gelfand-Cetlin fibers. In particular, we prove that any fiber over an interior point of an r-dimensional face of the Gelfand-Cetlin polytope is an isotropic submanifold and is diffeomorphic to Tr×N for some smooth manifold N and Tr≅(S1)r. We also prove that such N's are exactly the vanishing cycles shrinking to points in the associated toric variety via the toric degeneration. We also devise an algorithm of reading off Lagrangian fibers from the combinatorics of the ladder diagram.

LAGRANGIAN FIBERS OF GELFAND–CETLIN SYSTEMS OF SO(n)-TYPE

In this paper, we study the Gelfand–Cetlin systems and polytopes of the co-adjoint SO(n)-orbits. We describe the face structure of Gelfand–Cetlin polytopes and iterated bundle structure of Gelfand–Cetlin fibers in terms of combinatorics on the ladder diagrams. Using this description, we classify all Lagrangian fibers.

Monotone Lagrangians in Flag Varieties

Abstract In this paper, we give a formula for the Maslov index of a gradient holomorphic disk, which is a relative version of the Chern number formula of a gradient holomorphic sphere for a Hamiltonian $S^1$-action. Using the formula, we classify all monotone Lagrangian fibers of Gelfand–Cetlin systems on partial flag manifolds.

Singular fibres of the Gelfand-Cetlin system on 𝔲(n).

In this paper, we show that every singular fibre of the Gelfand-Cetlin system on co-adjoint orbits of unitary groups is a smooth isotropic submanifold which is diffeomorphic to a two-stage quotient of a compact Lie group by free actions of two other compact Lie groups. In many cases, these singular fibres can be shown to be homogeneous spaces or even diffeomorphic to compact Lie groups. We also give a combinatorial formula for computing the dimensions of all singular fibres, and give a detailed description of these singular fibres in many cases, including the so-called (multi-)diamond singularities. These (multi-)diamond singular fibres are degenerate for the Gelfand-Cetlin system, but they are Lagrangian submanifolds diffeomorphic to direct products of special unitary groups and tori. Our methods of study are based on different ideas involving complex ellipsoids, Lie groupoids and also general ideas coming from the theory of singularities of integrable Hamiltonian systems.This article is part of the theme issue 'Finite dimensional integrable systems: new trends and methods'.

Floer cohomologies of non-torus fibers of the Gelfand–Cetlin system

The Gelfand-Cetlin system has non-torus Lagrangian fibers on some of the boundary strata of the moment polytope. We compute Floer cohomologies of such non-torus Lagrangian fibers in the cases of the 3-dimensional full flag manifold and the Grassmannian of 2-planes in a 4-space.

Geometric Quantization of real polarizations via sheaves

In this article we develop tools to compute the Geometric Quantization of a symplectic manifold with respect to a regular Lagrangian foliation via sheaf cohomology and obtain important new applications in the case of real polarizations. The starting point is the definition of representation spaces due to Kostant. Besides the classical examples of Gelfand-Cetlin systems due to Guillemin and Sternberg very few examples of explicit computations of real polarizations are known. The computation of Geometric Quantization for Gelfand-Cetlin systems is based on a theorem due to \'Sniatycki for fibrations which identifies the representation space with the set of Bohr-Sommerfeld leaves determined by the integral action coordinates. In this article we check that the associated sheaf cohomology apparatus of Geometric Quantization satisfies Mayer-Vietoris and K\"unneth formulae. As a consequence, a new short proof of this classical result for fibrations due to \'Sniatycki is obtained. We also compute Geometric Quantization with respect to any generic regular Lagrangian foliation on a 2-torus and the case of the irrational flow. In the way, we recover some classical results in the computation of foliated cohomology of these polarizations.

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.

Some remarks on the Gelfand Cetlin system

In the first section of this paper, we show that the functions in involution of the Gelfand–Cetlin system can be obtained from a λ-parametric Lax equation. In the second section, we observe that the Gelfand–Cetlin system has no obstructions to global action–angle coordinates, and we give an explicit expression of global (action) angle coordinates. In the third section, we remark the fact that the Gelfand–Cetlin system is obtained via a nesting of superintegrable systems, and show they all present a non-vanishing Chern class.

The Gelfand-Cetlin system and quantization of the complex flag manifolds

The construction of angle action variables for collective completely integrable systems is described and the associated Bohr-Sommerfeld sets are determined. The quantization method of Sniatycki applied to such systems gives formulas for multiplicities. For the Gelfand-Cetlin system on complex flag manifolds we show that these formulas give the correct answers for the multiplicities of the associated representations.