Complex Flag Manifolds Research Articles

REDUCTIVE COMPACT HOMOGENEOUS CR MANIFOLDS

We define and investigate a class of compact homogeneous CR manifolds, that we call $$ \mathfrak{n} $$ -reductive. They are orbits of minimal dimension of a compact Lie group K 0 in algebraic affine homogeneous spaces of its complexification K. For these manifolds we obtain canonical equivariant fibrations onto complex flag manifolds, generalizing the Hopf fibration $$ {S^3}\to \mathbb{C}{{\mathbb{P}}^1} $$ . These fibrations are not, in general, CR submersions, but satisfy the weaker condition of being CR-deployments; to obtain CR submersions we need to strengthen their CR structure by lifting the complex stucture of the base.

An explicit retraction of symplectic flag manifolds onto complex flag manifolds

We construct an explicit deformation retraction of the manifold of symplectic flags onto the manifold of complex flags. The main tool is the polar decomposition of symplectic matrices. We also give a new definition of symplectic Stiefel manifold and prove that it has the same homotopy type as the complex Stiefel manifold.

Cycle space constructions for exhaustions of flag domains

In the study of complex flag manifolds, flag domains and their cycle spaces, a key point is the fact that the cycle space $\mathcal M_D$ of a flag domain $D$ is a Stein manifold. That fact has a long history. The earliest approach relied on construction of a strictly plurisubharmonic function on $\mathcal M_D$, starting with a $q$--convex exhaustion function on $D$, where $q$ is the dimension of a particular maximal compact subvariety of $D$ (we use the normalization that 0--convex means Stein). Construction of that exhaustion function on $D$ required that $D$ be measurable. In that case the exhaustion on $D$ was transferred to $\mathcal M_D$, using a special case of a method due to Barlet. Here we do the opposite: we use an incidence method to construct a canonical plurisubharmonic exhaustion function on $\mathcal M_D$ and use it in turn to construct a canonical $q$--convex exhaustion function on $D$. This promises to have strong consequences for cohomology vanishing theorems and the construction of admissible representations of real reductive Lie groups.

Orbits of real forms in complex flag manifolds

The authors study the structure and the CR geometry of the orbits $M$ of a real form $G_0$ of a complex semisimple Lie group $G$ in a complex flag manifold $X = G/Q$. It is shown that any such orbit $M$ has a tower of fibrations over a canonically associated real flag manifold $M_e$ with fibers that are products of Euclidean complex spaces and open orbits in complex flag manifolds. This result is used to investigate some topological properties of $M$. For example, it is proved that the fundamental group $\pi_1(M)$ depends only on $M_e$ and on the conjugacy class of the maximally noncompact Cartan subgroups of the isotropy of the action of $G_0$ on $M$. In particular, the fundamental group of a closed orbit $M$ is isomorphic to that of $M_e$. Many other deep results about properties of the CR structure of the orbits and its invariants and about $G_0$-equivarant maps between orbits are obtained.

Classical Symmetries of Complex Manifolds

We consider complex manifolds that admit actions by holomorphic transformations of classical simple real Lie groups and classify all such manifolds in a natural situation. Under our assumptions, which require the group at hand to be dimension-theoretically large with respect to the manifold on which it is acting, our classification result states that the manifolds which arise are described precisely as invariant open subsets of certain complex flag manifolds associated to the complexified groups.

Controllability and simultaneous controllability of isospectral bilinear control systems on complex flag manifolds

For isospectral bilinear control systems evolving on the so-called complex flag manifolds (i.e., on the orbits of the Hermitian matrices under unitary conjugation action) it is shown that controllability is almost always verified. Easy and generic sufficient conditions are provided. The result applies to the problem of density operator controllability of finite dimensional quantum mechanical systems. In addition, we show that systems having different drifts (corresponding for example to different Larmor frequencies) are simultaneously controllable by the same control field.

Gauging spacetime symmetries on the worldsheet and the geometric Langlands program — II

We study the two-dimensional twisted (0, 2) sigma-model on various smooth complex flag manifolds G/B, and explore its relevance to the geometric Langlands program. We find that an equivalence—at the level of the holomorphic chiral algebra—between a bosonic string on G/B and a B-gauged version of itself on G, will imply an isomorphism of classical -algebras and a level relation which underlie a geometric Langlands correspondence for G = SL(N, C). This furnishes an alternative physical interpretation of the geometric Langlands correspondence for G = SL(N, C), to that demonstrated earlier by Kapustin and Witten via an electric-magnetic duality of four-dimensional gauge theory. Likewise, the Hecke operators and Hecke eigensheaves will have an alternative physical interpretation in terms of the correlation functions of local operators in the holomorphic chiral algebra of a quasi-topological sigma-model without boundaries. A forthcoming paper will investigate the interpretation of a ``quantum'' geometric Langlands correspondence for G = SL(N, C) in a similar setting, albeit with fluxes of the sigma-model moduli which induce a ``quantum'' deformation of the relevant classical algebras turned on.

An extension of the (1,2)-symplectic property for -structures on flag manifolds

The (1,1)-symplectic property for -structures on a complex Riemannian manifold is a natural extension of the (1,2)-symplectic property for almost-complex structures on , and arises in the analysis of complex harmonic maps with values in . A characterization of this property in combinatorial terms is known only for almost-complex structures or when is the classical flag manifold . In this paper, we remove these restrictions by considering an intersection graph defined in terms of the corresponding root system. We prove that the -structure is (1,1)-symplectic exactly when the intersection graph is locally transitive. Our intersection graph construction may be helpful in characterizing many other Kähler-like properties on complex flag manifolds.

Chains in the Bruhat order

We study a family of polynomials whose values express degrees of Schubert varieties in the generalized complex flag manifold G/B. The polynomials are given by weighted sums over saturated chains in the Bruhat order. We derive several explicit formulas for these polynomials, and investigate their relations with Schubert polynomials, harmonic polynomials, Demazure characters, and generalized Littlewood-Richardson coefficients. In the second half of the paper, we study the classical flag manifold and discuss related combinatorial objects: flagged Schur polynomials, 312-avoiding permutations, generalized Gelfand-Tsetlin polytopes, the inverse Schubert-Kostka matrix, parking functions, and binary trees.

Gauging spacetime symmetries on the worldsheet and the geometric Langlands program

We study the two-dimensional twisted (0,2) sigma-model on various smooth complex flag manifolds G/B, and explore its relevance to the geometric Langlands program. We find that an equivalence - at the level of the holomorphic chiral algebra - between a bosonic string on G/B and a B-gauged version of itself on G, will imply an isomorphism of classical W-algebras and a level relation which underlie a geometric Langlands correspondence for G=SL(N,C). This furnishes an alternative physical interpretation of the geometric Langlands correspondence for G=SL(N,C), to that demonstrated earlier by Kapustin and Witten via an electric-magnetic duality of four-dimensional gauge theory. Likewise, the Hecke operators and Hecke eigensheaves will have an alternative physical interpretation in terms of the correlation functions of local operators in the holomorphic chiral algebra of a quasi-topological sigma-model without boundaries. A forthcoming paper will investigate the interpretation of a ``quantum'' geometric Langlands correspondence for G=SL(N,C) in a similar setting, albeit with fluxes of the sigma-model moduli which induce a ``quantum'' deformation of the relevant classical algebras turned on.

Twistors and 3-symmetric spaces

We describe complex twistor spaces over inner 3-symmetric spaces G/H, such that H acts transitively on the fibre. As in the symmetric case, the complex twistor spaces are flag manifolds G/K, where K is the centralizer of a torus in G. Moreover, they carry an almost complex structure defined using the horizontal distribution of the normal connection on G/H that coincides with the complex structure associated to a parabolic subgroup P⊂Gℂ if it is integrable. Conversely, starting from a complex flag manifold Gℂ/P, there exists a natural fibration with complex fibres on a 3-symmetric space, called fibration of degree 3.

On the topology of minimal orbits in complex flag manifolds

We compute the Euler-Poincar\'e characteristic of the homogeneous compact manifolds that can be described as minimal orbits for the action of a real form in a complex flag manifold.

Coupling Nonlinear Sigma-Matter to Yang-Mills Fields: Symmetry Breaking Patterns

We extend the traditional formulation of Gauge Field Theory by incorporating the (non-Abelian) gauge group parameters (traditionally simple spectators) as new dynamical (nonlinear-sigma-model-type) fields. These new fields interact with the usual Yang-Mills fields through a generalized minimal coupling prescription, which resembles the so-called Stueckelberg transformation [1], but for the non-Abelian case. Here we study the case of internal gauge symmetry groups, in particular, unitary groups U(N). We show how to couple standard Yang-Mills Theory to Nonlinear-Sigma Models on cosets of U(N): complex projective, Grassman and flag manifolds. These different couplings lead to distinct (chiral) symmetry breaking patterns and Higgs-less mass-generating mechanisms for Yang-Mills fields.

Feedback Stabilization of Isospectral Control Systems on Complex Flag Manifolds: Application to Quantum Ensembles

In an N-level quantum mechanical system, the problem of unitary feedback stabilization of mixed density operators to periodic orbits admits a natural Lyapunov-based time-varying feedback design. A global description of the domain of attraction of the closed-loop system can be provided based on a ``root-space''-like structure of the space of density operators. This convex set foliates as a complex flag manifold where each leaf is identified with the coadjoint orbit of the eigenvalues of the density operator. The converging conditions are time-independent but depend from the topology of the flag manifold: it is shown that the closed loop must have a number of equilibria at least equal to the Euler characteristic of the manifold, thus imposing obstructions of topological nature to global stabilizability.

CONTROLLABILITY OF ISOSPECTRAL BILINEAR CONTROL SYSTEMS ON COMPLEX FLAG MANIFOLDS

For isospectral bilinear control systems evolving on the so-called complex flag manifolds (i.e., on the orbits of the Hermitian matrices under unitary conjugation action) it is shown that controllability is almost always verified. Easy and generic sufficient conditions are provided. The result applies to the problem of density operator controllability of finite dimensional quantum mechanical systems.

STEEPEST DESCENT ON REAL FLAG MANIFOLDS

Real flag manifolds are the isotropy orbits of noncompact symmetric spaces G/K. Any such manifold M is acted on transitively by the (noncompact) Lie group G, and it is embedded in euclidean space as a taut submanifold. The aim of this paper is to show that the gradient flow of any height function is a one-parameter subgroup of G, where the gradient is defined with respect to a suitable homogeneous metric s on M; this generalizes the Kähler metric on adjoint orbits (the so-called complex flag manifolds). 2000 Mathematics Subject Classification 53C30, 53C35.

Poisson structures on complex flag manifolds associated with real forms

For a complex semisimple Lie group $G$ and a real form $G_0$ we define a Poisson structure on the variety of Borel subgroups of $G$ with the property that all $G_0$-orbits in $X$ as well as all Bruhat cells (for a suitable choice of a Borel subgroup of $G$) are Poisson submanifolds. In particular, we show that every non-empty intersection of a $G_0$-orbit and a Bruhat cell is a regular Poisson manifold, and we compute the dimension of its symplectic leaves.

Integrability of Analytic Almost Complex Structures on Banach Manifolds

We prove that the classical integrability condition for almost complex structures on finite-dimensional smooth manifolds also works in infinite dimensions in the case of almost complex structures that are real analytic on real analytic Banach manifolds. With this result at hand, we extend some known results concerning existence of invariant complex structures on homogeneous spaces of Banach–Lie groups. By way of illustration, we construct the complex flag manifolds associated with unital C*-algebras.

GROMOV–WITTEN INVARIANTS OF FLAG MANIFOLDS, VIA D-MODULES

We present a method for computing the 3-point genus zero Gromov-Witten invariants of the complex flag manifold from the relations of the small quantum cohomology algebra ( is a complex semisimple Lie group and is a Borel subgroup). In [3] and [9], at least in the case , two algebraic/combinatoric methods have been proposed, based on suitably designed axioms. Our method is quite different, being differential geometric in nature; it is based on the approach to quantum cohomology described in [7], which is in turn based on the integrable systems point of view of Dubrovin and Givental.

A Poisson Structure on Compact Symmetric Spaces

We present some basic results on a natural Poisson structure on any compact symmetric space. The symplectic leaves of this structure are related to the orbits of the corresponding real semisimple group on the complex flag manifold.