Hermitian Symmetric Spaces Research Articles

Plücker coordinates and the Rosenfeld planes

The exceptional compact hermitian symmetric space EIII is the quotient E6/Spin(10)×Z4U(1). We introduce the Plücker coordinates which give an embedding of EIII into CP26 as a projective subvariety. The subvariety is cut out by 27 Plücker relations. We show that, using Clifford algebra, one can solve this over-determined system of relations, giving local coordinate charts to the space.Our motivation is to understand EIII as the complex projective octonion plane (C⊗O)P2, whose construction is somewhat scattered across the literature. We will see that the EIII has an atlas whose transition functions have clear octonion interpretations, apart from those covering a sub-variety X∞ of dimension 10. This subvariety is itself a hermitian symmetric space known as DIII, with no apparent octonion interpretation. We give detailed analysis of the geometry in the neighbourhood of X∞.We further decompose X=EIII into F4-orbits: X=Y0∪Y∞, where Y0∼(OP2)C is an open F4-orbit and is the complexification of OP2, whereas Y∞ has co-dimension 1, thus EIII could be more appropriately denoted as (OP2)C‾. This decomposition appears in the classification of equivariant completion of homogeneous algebraic varieties by Ahiezer [2].

Localized waves on the periodic background for the Hermitian symmetric space derivative nonlinear Schrödinger equation

In this letter, we further investigate the Hermitian symmetric space derivative nonlinear Schrödinger equation through the development of a semi-degenerate Darboux transformation. To demonstrate the utility of this approach, we first reveal the expression of the double-periodic wave solution. On the periodic background, we visualize the kink-breather wave, the rogue wave and their interaction.

Long-time asymptotic behavior for the Hermitian symmetric space derivative nonlinear Schrödinger equation

Abstract Resorting to the spectral analysis of the 4 × 4 matrix spectral problem, we construct a 4 × 4 matrix Riemann–Hilbert problem to solve the initial value problem for the Hermitian symmetric space derivative nonlinear Schrödinger equation. The nonlinear steepest decent method is extended to study the 4 × 4 matrix Riemann–Hilbert problem, from which the various Deift–Zhou contour deformations and the motivation behind them are given. Through some proper transformations between the corresponding Riemann–Hilbert problems, the basic Riemann–Hilbert problem is reduced to a model Riemann–Hilbert problem, by which the long-time asymptotic behavior to the solution of the initial value problem for the Hermitian symmetric space derivative nonlinear Schrödinger equation is obtained with the help of the asymptotic expansion of the parabolic cylinder function and strict error estimates.

Pseudoeffective thresholds and cohomology of twisted symmetric tensor fields on irreducible Hermitian symmetric spaces

Pseudoeffective thresholds and cohomology of twisted symmetric tensor fields on irreducible Hermitian symmetric spaces

Cartan–Helgason theorem for quaternionic symmetric and twistor spaces

Let (g,k) be a complex quaternionic symmetric pair with k having an ideal sl(2,ℂ), k=sl(2,ℂ)+mc. Consider the representation Sm(ℂ2)=ℂm+1 of k via the projection k→sl(2,ℂ) onto the ideal sl(2,ℂ). We study the finite dimensional irreducible representations V(λ) of g which contain Sm(ℂ2) under k⊆g. We give a characterization of all such representations V(λ) and find the corresponding multiplicity, the dimension of Hom(V(λ)|k,Sm(ℂ2)). We consider also the branching problem of V(λ) under l=u(1)ℂ+mc⊂k and find the multiplicities. Geometrically the Lie subalgebra l⊂k defines a twistor space over the compact symmetric space of the compact real form Gu of Gℂ, Lie(Gℂ)=g, and our results give the decomposition for the L2-spaces of sections of certain vector bundles over the symmetric space and line bundles over the twistor space. This generalizes Cartan–Helgason’s theorem for symmetric spaces (g,k) and Schlichtkrull’s theorem for Hermitian symmetric spaces where one-dimensional representations of k are considered.

Birational Transformations on Irreducible Compact Hermitian Symmetric Spaces

Abstract We construct a sequence of explicit blow-ups and blow-downs on an irreducible compact Hermitian symmetric spaces $X$ which transforms it into a projective space of the same dimension. Moreover, this resolves a birational map given by Landsberg and Manivel. Centers of the blow-ups for $X$ are constructed by loci of chains of minimal rational curves and centers of the blow-ups for the projective space are constructed from the variety of minimal rational tangents of $X$ and its higher secant varieties. The result was known in the special case where $X$ is of rank 2 and could be found in Zak’s monograph “Tangents and secants of algebraic varieties.”

Geometry of Hermitian symmetric spaces under the action of a maximal unipotent group

Let [Formula: see text] be a non-compact irreducible Hermitian symmetric space of rank [Formula: see text] and let [Formula: see text] be an Iwasawa decomposition of [Formula: see text]. The group [Formula: see text] acts on [Formula: see text] by biholomorphisms and the real [Formula: see text]-dimensional submanifold [Formula: see text] intersects every [Formula: see text]-orbit transversally in a single point. Moreover [Formula: see text] is contained in a complex [Formula: see text]-dimensional submanifold of [Formula: see text] biholomorphic to [Formula: see text], the product of [Formula: see text] copies of the upper half-plane in [Formula: see text]. This fact leads to a one-to-one correspondence between [Formula: see text]-invariant domains in [Formula: see text] and tube domains in [Formula: see text]. In this setting we prove an analogue of Bochner’s tube theorem. Namely, an [Formula: see text]-invariant domain [Formula: see text] in [Formula: see text] is Stein if and only if the base [Formula: see text] of the associated tube domain is convex and “cone invariant”. We also prove the univalence of [Formula: see text]-invariant holomorphically separable Riemann domains over [Formula: see text]. This yields a precise description of the envelope of holomorphy of an arbitrary [Formula: see text]-invariant domain in [Formula: see text]. Finally, we obtain a characterization of several classes of [Formula: see text]-invariant plurisubharmonic functions on [Formula: see text] in terms of the corresponding classes of convex functions on [Formula: see text]. As an application we present an explicit Lie group theoretical description of all [Formula: see text]-invariant potentials of the Killing metric on [Formula: see text] and of the associated moment maps.

Moment maps and isoparametric hypersurfaces in spheres — Grassmannian cases

We expect that every Cartan–Münzner polynomial of degree four can be described as a squared-norm of a moment map for a Hamiltonian action. Our expectation is known to be true for Hermitian cases, that is, those obtained from the isotropy representations of compact irreducible Hermitian symmetric spaces of rank two. In this paper, we prove that our expectation is true for the Cartan–Münzner polynomials obtained from the isotropy representations of Grassmannian manifolds of rank two over R, C or H. The quaternion cases are the first non-Hermitian examples that our expectation is verified.

Weighted Bergman kernels for nearly holomorphic functions on bounded symmetric domains

We identify the standard weighted Bergman kernels of spaces of nearly holomorphic functions, in the sense of Shimura, on bounded symmetric domains. This also yields a description of the analogous kernels for spaces of “invariantly-polyanalytic” functions — a generalization of the ordinary polyanalytic functions on the ball which seems to be the most appropriate one from the point of view of holomorphic invariance. In both cases, the kernels turn out to be given by certain spherical functions, or equivalently Heckman-Opdam hypergeometric functions, and a conjecture relating some of these to a Faraut-Koranyi hypergeometric function is formulated based on the study of low rank situations. Finally, analogous results are established also for compact Hermitian symmetric spaces, where explicit formulas in terms of multivariable Jacobi polynomials are given.

Uniformizations of Compact Sasakian Manifolds

Abstract We give a criterion for compact Sasakian manifolds to be deformed to Sasakian manifolds, which are locally isomorphic to circle bundles of anti-canonical bundles over Hermitian symmetric spaces as a Sasakian analogue of Simpson’s uniformization results related to variations of Hodge structure and Higgs bundles.

Spherical CR-symmetric hypersurfaces in Hermitian symmetric spaces

Spherical CR-symmetric hypersurfaces in Hermitian symmetric spaces

Darboux transformation for nonlinear Schrödinger type hierarchies

The Darboux transformation for the nonlinear Schrödinger type hierarchies associated to the Hermitian symmetric spaces can be gotten by using the factorization theory of corresponding group. Based on this, we give the n-fold Darboux transformations in determinant representation, and some explicit soliton solutions are constructed.

Effective actions in supersymmetric gauge theories: heat kernels for non-minimal operators

We study the quantum dynamics of a system of n Abelian mathcal{N} = 1 vector multiplets coupled to frac{1}{2}nleft(n+1right) chiral multiplets which parametrise the Hermitian symmetric space Sp(2n, ℝ)/U(n). In the presence of supergravity, this model is super-Weyl invariant and possesses the maximal non-compact duality group Sp(2n, ℝ) at the classical level. These symmetries should be respected by the logarithmically divergent term (the “induced action”) of the effective action obtained by integrating out the vector multiplets. In computing the effective action, one has to deal with non-minimal operators for which the known heat kernel techniques are not directly applicable, even in flat (super)space. In this paper we develop a method to compute the induced action in Minkowski superspace. The induced action is derived in closed form and has a simple structure. It is a higher-derivative superconformal sigma model on Sp(2n, ℝ)/U(n). The obtained mathcal{N} = 1 results are generalised to the case of mathcal{N} = 2 local supersymmetry: a system of n Abelian mathcal{N} = 2 vector multiplets coupled to mathcal{N} = 2 chiral multiplets XI parametrising Sp(2n, ℝ)/U(n). The induced action is shown to be proportional to int {textrm{d}}^4x{textrm{d}}^4theta {textrm{d}}^4overline{theta}Emathfrak{K}left(X,overline{X}right) , where mathfrak{K}left(X,overline{X}right) is the Kähler potential for Sp(2n, ℝ)/U(n). We also apply our method to compute DeWitt’s a2 coefficients in some non-supersymmetric theories with non-minimal operators.

On gap rigidity problems for compact Hermitian symmetric spaces

We prove a gap rigidity theorem for diagonal curves in maximal polyspheres of irreducible compact Hermitian symmetric spaces of tube type, which is a dual analog to a theorem obtained by Mok. Motivated by the proof we show that the Chow space of certain totally geodesic submanifolds is affine algebraic, which gives a weaker version of gap rigidity for a class of higher dimensional submanifolds.

Generalized Darboux transformation for nonlinear Schrödinger system on general Hermitian symmetric spaces and rogue wave solutions

In this paper, a generalized Darboux transformation is obtained for Fordy–Kulish NLS (nonlinear Schrödinger) systems on general Hermitian symmetric spaces in order to rigorously obtain rogue wave solutions for these systems. In particular, we express the generalized algebraic relations in a simple and elegant compact form. As an illustration, we derive multi-soliton, breather-type and mainly rogue wave solutions of triangular patterns for single- and multi-component NLS systems on [Formula: see text] and [Formula: see text] respectively. We also analyze the modulation instability of proper plane wave solutions. In order to get visual intuition for the dynamics of the result and solutions for the running examples, the associated simulations of profiles are furnished as well.

Weyl invariance, non-compact duality and conformal higher-derivative sigma models

We study a system of n Abelian vector fields coupled to frac{1}{2} n(n+1) complex scalars parametrising the Hermitian symmetric space {textsf{Sp}}(2n, {mathbb {R}})/ {textsf{U}}(n). This model is Weyl invariant and possesses the maximal non-compact duality group {textsf{Sp}}(2n, {mathbb {R}}). Although both symmetries are anomalous in the quantum theory, they should be respected by the logarithmic divergent term (the “induced action”) of the effective action obtained by integrating out the vector fields. We compute this induced action and demonstrate its Weyl and {textsf{Sp}}(2n, {mathbb {R}}) invariance. The resulting conformal higher-derivative sigma -model on {textsf{Sp}}(2n, {mathbb {R}})/ {textsf{U}}(n) is generalised to the cases where the fields take their values in (i) an arbitrary Kähler space; and (ii) an arbitrary Riemannian manifold. In both cases, the sigma -model Lagrangian generates a Weyl anomaly satisfying the Wess–Zumino consistency condition.

Comparison of quantizations of symmetric spaces: cyclotomic Knizhnik–Zamolodchikov equations and Letzter–Kolb coideals

Abstract We establish an equivalence between two approaches to quantization of irreducible symmetric spaces of compact type within the framework of quasi-coactions, one based on the Enriquez–Etingof cyclotomic Knizhnik–Zamolodchikov (KZ) equations and the other on the Letzter–Kolb coideals. This equivalence can be upgraded to that of ribbon braided quasi-coactions, and then the associated reflection operators (K-matrices) become a tangible invariant of the quantization. As an application we obtain a Kohno–Drinfeld type theorem on type $\mathrm {B}$ braid group representations defined by the monodromy of KZ-equations and by the Balagović–Kolb universal K-matrices. The cases of Hermitian and non-Hermitian symmetric spaces are significantly different. In particular, in the latter case a quasi-coaction is essentially unique, while in the former we show that there is a one-parameter family of mutually nonequivalent quasi-coactions.

Non-split supermanifolds associated with the cotangent bundle

Here, I study the problem of classification of non-split supermanifolds having as retract the split supermanifold $(M,\Omega)$, where $\Omega$ is the sheaf of holomorphic forms on a given complex manifold $M$ of dimension $> 1$. I propose a general construction associating with any $d$-closed $(1,1)$-form $\omega$ on $M$ a supermanifold with retract $(M,\Omega)$ which is non-split whenever the Dolbeault class of $\omega$ is non-zero. In particular, this gives a non-empty family of non-split supermanifolds for any flag manifold $M\ne \mathbb{CP}^1$. In the case where $M$ is an irreducible compact Hermitian symmetric space, I get a complete classification of non-split supermanifolds with retract $(M,\Omega)$. For each of these supermanifolds, the 0- and 1-cohomology with values in the tangent sheaf are calculated. As an example, I study the $\Pi$-symmetric super-Grassmannians introduced by Yu. Manin.

Curvature Invariants of Equivariant Isometric Minimal Immersions into Grassmannian Manifolds

Using theories of principal fibre bundles and connections in [11], we study curvature invariants as Riemannian submanifolds for equivariant isometric minimal immersions from Riemannian or Hermitian symmetric spaces of compact type into Grassmannian manifolds. First we calculate principal curvatures of all equivariant full isometric minimal immersions from the complex projective line to complex quadrics. It is shown that they do not depend on the parameter of the moduli space. According to the preceding research, it is important for the study of curvature invariants to examine the behavior of higher order covariant derivative of the second fundamental form. It also enables us to distinguish those maps by curvature invariants in the moduli space. In addition, we calculate principal curvatures of all equivariant full holomorphic isometric embeddings between the complex projective spaces.

The Hermitian symmetric space Fokas–Lenells equation: spectral analysis and long-time asymptotics

Abstract Based on the inverse scattering transformation, we carry out spectral analysis of the $4\times 4$ matrix spectral problems related to the Hermitian symmetric space Fokas–Lenells (FL) equation, by which the solution of the Cauchy problem of the Hermitian symmetric space FL equation is transformed into the solution of a Riemann–Hilbert problem. The nonlinear steepest descent method is extended to study the Riemann–Hilbert problem, from which the various Deift–Zhou contour deformations and the motivation behind them are given. Through some proper transformations between the corresponding Riemann–Hilbert problems and strict error estimates, we obtain explicitly the long-time asymptotics of the Cauchy problem of the Hermitian symmetric space FL equation with the aid of the parabolic cylinder function.