Complex Flag Manifolds Research Articles

Point particle E-models

We show that the same algebraic data that permit to construct the Lax pair and the r-matrix of an integrable non-linear σ-model in 1 + 1 dimensions can be also used for the construction of Lax pairs and of r-matrices of several other non-trivial integrable theories in 1 + 0 dimension. We call those new integrable theories the point particle E-models, we describe their structure and give their physical interpretation. We work out in detail the point particle E-modelsassociated to the bi-Yang–Baxter deformation of the SU(N) principal chiral model. In particular, for each complex flag manifold we thus obtain a two-parameter family of integrable models living on it.

Hermitian non-K<span class="BxUVEf ILfuVd" lang="en"><span class="hgKElc">ä</span></span>hler structures on products of principal $\mathrm{S}^1$-bundles over complex flag manifolds and applications in Hermitian geometry with torsion

Hermitian non-K<span class="BxUVEf ILfuVd" lang="en"><span class="hgKElc">ä</span></span>hler structures on products of principal $\mathrm{S}^1$-bundles over complex flag manifolds and applications in Hermitian geometry with torsion

On finitely nondegenerate closed homogeneous CR manifolds

A complex flag manifold $$\textsf {F}{=}{{\textbf {G}}}/{{\textbf {Q}}}$$ decomposes into finitely many real orbits under the action of a real form $${{\textbf {G}}}^\upsigma $$ of $${{\textbf {G}}}$$ . Their embedding into $$\textsf {F}$$ defines on them CR manifold structures. We characterize and list all the closed real orbits which are finitely nondegenerate.

On self-maps of complex flag manifolds

On self-maps of complex flag manifolds

Cohomology of generalized Dold spaces

Let (X,J) be an almost complex manifold with a (smooth) involution σ:X→X such that Fix(σ)≠∅. Assume that σ is a complex conjugation, i.e., the differential of σ anti-commutes with J. The space P(m,X):=Sm×X/∼ where (v,x)∼(−v,σ(x)) was referred to as a generalized Dold manifold. The above definition admits an obvious generalization to a much wider class of spaces where X,S are arbitrary topological spaces. The resulting space P(S,X) will be called a generalized Dold space. When S and X are CW complexes satisfying certain natural requirements, we obtain a CW-structure on P(S,X). Under certain further hypotheses, we determine the mod 2 cohomology groups of P(S,X). We determine the Z2-cohomology algebra when X is (i) a torus manifold whose torus orbit space is a homology polytope, (ii) a complex flag manifold. One of the main tools is the Stiefel-Whitney class formula for vector bundles over P(S,X) associated to σ-conjugate complex bundles over X when the S is a paracompact Hausdorff topological space, extending the validity of the formula, obtained earlier by Nath and Sankaran, in the case of generalized Dold manifolds.

On Families of Wigner Functions for N-Level Quantum Systems

A method for constructing all admissible unitary non-equivalent Wigner quasiprobability distributions providing the Stratonovic-h-Weyl correspondence for an arbitrary N-level quantum system is proposed. The method is based on the reformulation of the Stratonovich–Weyl correspondence in the form of algebraic “master equations” for the spectrum of the Stratonovich–Weyl kernel. The later implements a map between the operators in the Hilbert space and the functions in the phase space identified by the complex flag manifold. The non-uniqueness of the solutions to the master equations leads to diversity among the Wigner quasiprobability distributions. It is shown that among all possible Stratonovich–Weyl kernels for a N=(2j+1)-level system, one can always identify the representative that realizes the so-called SU(2)-symmetric spin-j symbol correspondence. The method is exemplified by considering the Wigner functions of a single qubit and a single qutrit.

Higher order Levi forms on homogeneous CR manifolds

We investigate the nondegeneracy of higher order Levi forms on weakly nondegenerate homogeneous CR manifolds. Improving previous results, we prove that general orbits of real forms in complex flag manifolds have order less or equal than 3 and the compact ones less or equal 2. Finally we construct by Lie extensions weakly nondegenerate CR vector bundles with arbitrary orders of nondegeneracy.

A note on the equivariant cobordism of generalized Dold manifolds

Let (X,J) be an almost complex manifold with a (smooth) involution σ:X→X such that Fix(σ)≠∅. Assume that σ is a complex conjugation, i.e., the differential of σ anti-commutes with J. The space P(m,X):=Sm×X/∼ where (v,x)∼(−v,σ(x)) is known as a generalized Dold manifold. Suppose that a group G≅Z2s acts smoothly on X such that g∘σ=σ∘g for all g∈G. Using the action of the diagonal subgroup D=O(1)m+1⊂O(m+1) on the sphere Sm, we obtain an action of G=D×G on Sm×X, which descends to a (smooth) action of G on P(m,X). When the stationary point set XG for the G action on X is finite, the same also holds for the G action on P(m,X). The main result of this note is that the equivariant cobordism class [P(m,X),G] vanishes if and only if [X,G] vanishes. We illustrate this result in the case when X is the complex flag manifold, σ is the natural complex conjugation and G≅(Z2)n is contained in the diagonal subgroup of U(n).

Infinitesimally tight Lagrangian orbits

We describe isotropic orbits for the restricted action of a subgroup of a Lie group acting on a symplectic manifold by Hamiltonian symplectomorphisms and admitting an Ad*-equivariant moment map. We obtain examples of Lagrangian orbits of complex flag manifolds, of cotangent bundles of orthogonal Lie groups, and of products of flags. We introduce the notion of infinitesimally tight and study the intersection theory of such Lagrangian orbits, giving many examples.

Cohomology rings of the real and oriented partial flag manifolds

We show, in contrast to the cases of complex and quaternionic partial flag manifolds, the rational cohomology rings of the real and oriented partial flag manifolds are generated not only by the characteristic classes of their canonical vector bundles, but also by exterior algebras arising from the rational cohomology rings of real Stiefel manifolds. We also discuss the rational and the mod-2 equivariant cohomology rings of these partial flag manifolds for certain canonical actions of tori and 2-tori.

Gröbner bases for (partial) flag manifolds

For an arbitrary complex (partial) flag manifold F, a Gröbner basis for the ideal which (in the Borel picture) determines the cohomology algebra H⁎(F;Z) is obtained. This Gröbner basis is used to derive multiplication rules for a convenient additive basis of H⁎(F;Z) given in terms of Chern classes of the canonical bundles over F. The analogous Gröbner bases related to the mod 2 cohomology of real flag manifolds are also presented.

Calabi-Yau metrics on canonical bundles of complex flag manifolds

In the present paper we provide a description of complete Calabi-Yau metrics on the canonical bundle of generalized complex flag manifolds. By means of Lie theory we give an explicit description of complete Ricci-flat Kähler metrics obtained through the Calabi Ansatz technique. We use this approach to provide several explicit examples of noncompact complete Calabi-Yau manifolds, these examples include canonical bundles of non-toric flag manifolds (e.g. Grassmann manifolds and full flag manifolds).

Homogeneous Contact Manifolds and Resolutions of Calabi–Yau Cones

In the present work we provide a constructive method to describe contact structures on compact homogeneous contact manifolds. The main feature of our approach is to describe the Cartan-Ehresmann connection (gauge field) for principal circle bundles over complex flag manifolds by using elements of representation theory of simple Lie algebras. This description allows us to compute explicitly the expression of the contact form for any Boothby-Wang fibration over complex flag manifolds as well as their induced homogeneous Sasaki-Einstein structures. As an application of our results we use the Cartan-Remmert reduction and the Calabi ansatz technique to provide many explicit examples of crepant resolutions of Calabi-Yau cones with certain homogeneous Sasaki-Einstein manifolds realized as links of isolated hypersurface singularities.

On generalized Dold manifolds

Let $X$ be a smooth manifold with a (smooth) involution $\sigma:X\to X$ such that ${\rm Fix}(\sigma)\ne \emptyset$. We call the space $P(m,X):=\mathbb{S}^m\times X/\!\sim$ where $(v,x)\sim (-v,\sigma(x))$ a generalized Dold manifold. When $X$ is an almost complex manifold and the differential $T\sigma: TX\to TX$ is conjugate complex linear on each fibre, we obtain a formula for the Stiefel-Whitney polynomial of $P(m,X)$ when $H^1(X;\mathbb{Z}_2)=0$. We obtain results on stable parallelizability of $P(m,X)$ and a very general criterion for the (non) vanishing of the unoriented cobordism class $[P(m,X)]$ in terms of the corresponding properties for $X$. These results are applied to the case when $X$ is a complex flag manifold.

On the intersection pairing between cycles in SU(p,q)-flag domains and maximally real Schubert varieties

An SU(p,q)-flag domain is an open orbit of the real Lie group SU(p,q) acting on the complex flag manifold associated to its complexification SL(p+q,C). Any such flag domain contains certain compact complex submanifolds, called cycles, which encode much of the topological, complex geometric and representation theoretical properties of the flag domain. This article is concerned with the description of these cycles in homology using a specific type of Schubert varieties. They are defined by the condition that the fixed point of the Borel group in question is in the closed SU(p,q)-orbit in the ambient manifold. Equivalently, the Borel group contains the AN-factor of some Iwasawa decomposition. We consider the Schubert varieties of this type which are of complementary dimension to the cycles. It is known that if such a variety has non-empty intersection with a certain base cycle, then it does so transversally (in finitely many points). With the goal of understanding this duality, we describe these points of intersection in terms of flags as well as in terms of fixed points of a given maximal torus. The relevant Schubert varieties are described in terms of Weyl group elements. Much of our work is of an algorithmic nature, but, for example in the case of maximal parabolics, i.e. Grassmannians, formulas are derived.

Hilbert series of nearly holomorphic sections on generalized flag manifolds

Let X=G/P be a complex flag manifold and E→X be a G-homogeneous holomorphic vector bundle. Fix a U-invariant Kähler metric on X with U⊆G maximal compact. We study the sheaf of nearly holomorphic sections and show that the space of global nearly holomorphic sections in E coincides with the space of U-finite smooth sections in E. The degree of nearly holomorphic sections defines a U-invariant filtration on this space. Using sheaf cohomology, we determine in suitable cases the corresponding Hilbert series. It turns out that this is given in terms of Lusztig's q-analog of Kostant's weight multiplicity formula, and hence gives a new representation theoretic interpretation of these formulas.

Hamiltonian minimality of normal bundles over the isoparametric submanifolds

Let N be a complex flag manifold of a compact semi-simple Lie group G, which is standardly embedded in the Lie algebra g of G as a principal orbit of the adjoint action. We show that the normal bundle of N in g is a Hamiltonian minimal Lagrangian submanifold in the tangent space Tg which is naturally regarded as the complex Euclidean space. Moreover, we specify the complex flag manifolds with this property in the class of full irreducible isoparametric submanifolds in the Euclidean space.

Schur Times Schubert via the Fomin-Kirillov Algebra

We study multiplication of any Schubert polynomial $\mathfrak{S}_w$ by a Schur polynomial $s_{\lambda}$ (the Schubert polynomial of a Grassmannian permutation) and the expansion of this product in the ring of Schubert polynomials. We derive explicit nonnegative combinatorial expressions for the expansion coefficients for certain special partitions $\lambda$, including hooks and the $2\times 2$ box. We also prove combinatorially the existence of such nonnegative expansion when the Young diagram of $\lambda$ is a hook plus a box at the $(2,2)$ corner. We achieve this by evaluating Schubert polynomials at the Dunkl elements of the Fomin-Kirillov algebra and proving special cases of the nonnegativity conjecture of Fomin and Kirillov.This approach works in the more general setup of the (small) quantum cohomology ring of the complex flag manifold and the corresponding (3-point) Gromov-Witten invariants. We provide an algebro-combinatorial proof of the nonnegativity of the Gromov-Witten invariants in these cases, and present combinatorial expressions for these coefficients.

Convergence of Kähler to real polarizations on flag manifolds via toric degenerations

In this paper we construct a family of complex structures on a complex flag manifold that converge to the real polarization coming from the Gelfand-Cetlin integrable system, in the sense that holomorphic sections of a prequantum line bundle converge to delta-function sections supported on the Bohr-Sommerfeld fibers. Our construction is based on a toric degeneration of flag varieties and a deformation of K\"ahler structure on toric varieties by symplectic potentials.

The Geometry of Antiferromagnetic Spin Chains

It has been a conjecture of F.D.M.Haldane that long-wavelength excitations around antiferromagnetic vacua of certain spin chains are related to relativistic two-dimensional σ-models. We construct a class of such spin chains and associated σ-models, using symplectic geometry as a main tool. The target space of the σ-model can be an arbitrary complex flag manifold, and we find universal expressions for the metric and θ-term. The derivation relies on the fact that the flag manifold is a Lagrangian submanifold in a direct product of Grassmannians associated (via geometric quantization) to the representations of spins at consecutive sites of the spin chain. The true goal is not to reach the uttermost limits, but to discover a completeness that knows no boundaries. Rabindranath Tagore