Hermitian Symmetric Spaces Research Articles

Multicomponent Nonlinear Evolution Equations of the Heisenberg Ferromagnet Type: Local Versus Nonlocal Reductions

This work is dedicated to systems of matrix nonlinear evolution equations related to Hermitian symmetric spaces of the type $\mathbf{A.III}$. The systems under consideration generalize the $1+1$ dimensional Heisenberg ferromagnet equation in the sense that their Lax pairs are linear bundles in pole gauge like for the original Heisenberg model. Here we present certain local and nonlocal reductions. A local integrable deformation and some of its reductions are discussed as well.

Homogeneous Higgs and co-Higgs bundles on Hermitian symmetric spaces

We define homogeneous principal Higgs and co-Higgs bundles over irreducible Hermitian symmetric spaces of compact type. We provide a classification for each type of object up to isomorphism, which in each case can be interpreted as defining a moduli space.

Nearly holomorphic automorphic forms on SL2

We define the space of nearly holomorphic automorphic forms on a connected reductive group G over Q such that the homogeneous space G(R)1/K∞∘ is a Hermitian symmetric space. By Pitale, Saha and Schmidt's study, there are the classification of indecomposable (g,K∞)-modules which occur in the space of nearly holomorphic elliptic modular forms and Siegel modular forms of degree 2. This paper studies global representations of the adele group G(AQ) which occur in the space of nearly holomorphic Hilbert modular forms. In the case of elliptic modular forms, the result of this paper is an adelization of Pitale, Saha and Schmidt's result.

Compact Hermitian Symmetric Spaces, Coadjoint Orbits, and the Dynamical Stability of the Ricci Flow

Using a stability criterion due to Kröncke, we show, providing $${n\ne 2k}$$ , the Kähler–Einstein metric on the Grassmannian $$Gr_{k}(\mathbb {C}^{n})$$ of complex k-planes in an n-dimensional complex vector space is dynamically unstable as a fixed point of the Ricci flow. This generalises the recent results of Kröncke and Knopf–Sesum on the instability of the Fubini–Study metric on $$\mathbb {CP}^{n}$$ for $$n>1$$ . The key to the proof is using the description of Grassmannians as certain coadjoint orbits of SU(n). We are also able to prove that Kröncke’s method will not work on any of the other compact, irreducible, Hermitian symmetric spaces.

Supersymmetric nonlinear sigma models as anomalous gauge theories

We revisit supersymmetric nonlinear sigma models on the target manifold $CP^{N-1}$ and $SO(N)/SO(N-2)\times U(1)$ in four dimensions. These models are formulated as gauged linear models, but it is indicated that the Wess-Zumino term should be added to the linear model since the hidden local symmetry is anomalous. Applying a procedure used for quantization of anomalous gauge theories to the nonlinear models, we determine the form of the Wess-Zumino term, by which a global symmetry in the linear model becomes smaller in the action than the conventional one. Moreover, we analyze the resulting linear model in the $1/N$ leading order. Consequently, we find that the model has a critical coupling constant similar to bosonic models. In the weak coupling regime, the $U(1)$ local symmetry is broken but supersymmetry is never broken. In contrast to the bosonic case, it is impossible to find stable vacua in the strong coupling regime as far as in the $1/N$ leading order. These results are straightforwardly generalized to the case of the hermitian symmetric space.

Non-compact duality, super-Weyl invariance and effective actions

In both mathcal{N} = 1 and mathcal{N} = 2 supersymmetry, it is known that Sp(2n, ℝ) is the maximal duality group of n vector multiplets coupled to chiral scalar multiplets τ(x, θ) that parametrise the Hermitian symmetric space Sp(2n, ℝ)/U(n). If the coupling to τ is introduced for n superconformal gauge multiplets in a supergravity background, the action is also invariant under super-Weyl transformations. Computing the path integral over the gauge prepotentials in curved superspace leads to an effective action Γ[τ, overline{tau} ] with the following properties: (i) its logarithmically divergent part is invariant under super-Weyl and rigid Sp(2n, ℝ) transformations; (ii) the super-Weyl transformations are anomalous upon renormalisation. In this paper we describe the mathcal{N} = 1 and mathcal{N} = 2 locally supersymmetric “induced actions” which determine the logarithmically divergent parts of the corresponding effective actions. In the mathcal{N} = 1 case, superfield heat kernel techniques are used to compute the induced action of a single vector multiplet (n = 1) coupled to a chiral dilaton-axion multiplet. We also describe the general structure of mathcal{N} = 1 super-Weyl anomalies that contain weight-zero chiral scalar multiplets ΦI taking values in a Kähler manifold. Explicit anomaly calculations are carried out in the n = 1 case.

The Kähler geometry of certain optimal transport problems

Let $X$ and $Y$ be domains of $\mathbb{R}^n$ equipped with respective probability measures $\mu$ and $ \nu$. We consider the problem of optimal transport from $\mu$ to $\nu$ with respect to a cost function $c: X \times Y \to \mathbb{R}$. To ensure that the solution to this problem is smooth, it is necessary to make several assumptions about the structure of the domains and the cost function. In particular, Ma, Trudinger, and Wang established regularity estimates when the domains are strongly \textit{relatively $c$-convex} with respect to each other and cost function has non-negative \textit{MTW tensor}. For cost functions of the form $c(x,y)= \Psi(x-y)$ for some convex function $\Psi$, we find an associated K\"ahler manifold whose orthogonal anti-bisectional curvature is proportional to the MTW tensor. We also show that relative $c$-convexity geometrically corresponds to geodesic convexity with respect to a dual affine connection. Taken together, these results provide a geometric framework for optimal transport which is complementary to the pseudo-Riemannian theory of Kim and McCann. We provide several applications of this work. In particular, we find a complete K\"ahler surface with non-negative orthogonal bisectional curvature that is not a Hermitian symmetric space or biholomorphic to $\mathbb{C}^2$. We also address a question in mathematical finance raised by Pal and Wong on the regularity of \textit{pseudo-arbitrages}, or investment strategies which outperform the market.

Topological invariants of parabolic G-Higgs bundles

For a semisimple real Lie group $G$, we study topological properties of moduli spaces of polystable parabolic $G$-Higgs bundles over a Riemann surface with a divisor of finitely many distinct points. For a split real form of a complex simple Lie group, we compute the dimension of apparent parabolic Teichm{\"u}ller components. In the case of isometry groups of classical Hermitian symmetric spaces of tube type, we provide new topological invariants for maximal parabolic $G$-Higgs bundles arising from a correspondence to orbifold Higgs bundles. Using orbifold cohomology we count the least number of connected components of moduli spaces of such objects. We further exhibit an alternative explanation of fundamental results on counting components in the absence of a parabolic structure.

A characterization of complex space forms via Laplace operators

Inspired by the work of Lu and Tian (Duke Math J 125(2):351–387, 2004), in this paper we address the problem of studying those Kahler manifolds satisfying the $$\Delta$$ -property, i.e. such that on a neighborhood of each of its points the kth power of the Kahler Laplacian is a polynomial function of the complex Euclidean Laplacian, for all positive integer k (see below for its definition). We prove two results: (1) if a Kahler manifold satisfies the $$\Delta$$ -property then its curvature tensor is parallel; (2) if an Hermitian symmetric space of classical type satisfies the $$\Delta$$ -property then it is a complex space form (namely it has constant holomorphic sectional curvature). In view of these results we believe that if a Kahler manifold satisfies the $$\Delta$$ -property then it is a complex space form.

Contact hypersurfaces and CR-symmetry

We show that the contact $$(k,\mu )$$ -spaces whose Boeckx invariant is $$\ne 1$$ are realized as real hypersurfaces of the complex quadric $$Q^n$$ and its non-compact dual $$Q^{n*}$$ . Then, we classify simply connected, complete, non-K-contact, CR-symmetric contact metric spaces by real hypersurfaces in Hermitian symmetric spaces.

Embeddings from noncompact symmetric spaces to their compact duals

Every compact symmetric space $M$ admits a dual noncompact symmetric space $\check{M}$. When $M$ is a generalized Grassmannian, we can view $\check{M}$ as a open submanifold of it consisting of space-like subspaces \cite{HL}. Motivated from this, we study the embeddings from noncompact symmetric spaces to their compact duals, including space-like embedding for generalized Grassmannians, Borel embedding for Hermitian symmetric spaces and the generalized embedding for symmetric R-spaces. We will compare these embeddings and describe their images using cut loci.

A Dolbeault-Dirac Spectral Triple for Quantum Projective Space

The notion of a Kahler structure for a differential calculus was recently introduced by the second author as a framework in which to study the noncommutative geometry of the quantum flag manifolds. It was subsequently shown that any covariant positive definite Kahler structure has a canonically associated triple satisfying, up to the compact resolvent condition, Connes' axioms for a spectral triple. In this paper we begin the development of a robust framework in which to investigate the compact resolvent condition, and moreover, the general spectral behaviour of covariant Kahler structures. This framework is then applied to quantum projective space endowed with its Heckenberger-Kolb differential calculus. An even spectral triple with non-trivial associated K-homology class is produced, directly q-deforming the Dirac-Dolbeault operator of complex projective space. Finally, the extension of this approach to a certain canonical larger class of compact quantum Hermitian symmetric spaces is discussed in detail.

Cyclic parallel hypersurfaces in complex Grassmannians of rank 2

The object of the paper is to study cyclic parallel hypersurfaces in complex (hyperbolic) two-plane Grassmannians which have a remarkable geometric structure as Hermitian symmetric spaces of rank 2. First, we prove that if the Reeb vector field belongs to the orthogonal complement of the maximal quaternionic subbundle, then the shape operator of a cyclic parallel hypersurface in complex hyperbolic two-plane Grassmannians is Reeb parallel. By using this fact, we classify all cyclic parallel hypersurfaces in complex hyperbolic two-plane Grassmannians with non-vanishing geodesic Reeb flow. Next, we give a non-existence theorem for cyclic Hopf hypersurfaces in complex two-plane Grassmannians.

Volume-preserving mappings between Hermitian symmetric spaces of compact type

In this paper, we establish the rigidity result for local holomorphic volume preserving maps from an irreducible Hermitian symmetric space of compact type into its Cartesian products.

Ultralocal Lax connection for para-complex [formula omitted]-cosets

We consider σ-models on para-complex ZT-cosets, which are analogues of those on complex homogeneous target spaces considered recently by D. Bykov. For these models, we show the existence of a gauge-invariant Lax connection whose Poisson brackets are ultralocal. Furthermore, its light-cone components commute with one another in the sense of Poisson brackets. This extends a result of O. Brodbeck and M. Zagermann obtained twenty years ago for hermitian symmetric spaces.

Rigidity of certain admissible pairs of rational homogeneous spaces of Picard number 1 which are not of the subdiagram type

Recently, Mok and Zhang (2019) introduced the notion of admissible pairs (X0, X) of rational homogeneous spaces of Picard number 1 and proved rigidity of admissible pairs (X0, X) of the subdiagram type whenever X0 is nonlinear. It remains unsolved whether rigidity holds when (X0, X) is an admissible pair NOT of the subdiagram type of nonlinear irreducible Hermitian symmetric spaces such that (X0, X) is nondegenerate for substructures. In this article we provide sufficient conditions for confirming rigidity of such an admissible pair. In a nutshell our solution consists of an enhancement of the method of propagation of sub-VMRT (varieties of minimal rational tangents) structures along chains of minimal rational curves as is already implemented in the proof of the Thickening Lemma of Mok and Zhang (2019). There it was proven that, for a sub-VMRT structure $$\overline{\omega} : \mathscr{C}(S) \rightarrow S$$ on a uniruled projective manifold $$(X,\,{\cal K})$$ equipped with a minimal rational component and satisfying certain conditions so that in particular S is “uniruled” by open subsets of certain minimal rational curves on X, for a “good” minimal rational curve l emanating from a general point x ∈ S, there exists an immersed neighborhood Nl of l which is in some sense “uniruled” by minimal rational curves. By means of the Algebraicity Theorem of Mok and Zhang (2019), S can be completed to a projective subvariety Z ⊂ X. By the author’s solution of the Recognition Problem for irreducible Hermitian symmetric spaces of rank ⩾ 2 (2008) and under Condition (F), which symbolizes the fitting of sub-VMRTs into VMRTs, we further prove that Z is the image under a holomorphic immersion of X0 into X which induces an isomorphism on second homology groups. By studying ℂ*-actions we prove that Z can be deformed via a one-parameter family of automorphisms to converge to X0 ⊂ X. Under the additional hypothesis that all holomorphic sections in Γ(X0, Tx∣x0) lift to global holomorphic vector fields on X, we prove that the admissible pair (X0, X) is rigid. As examples we check that (X0, X) is rigid when X is the Grassmannian G(n, n) of n-dimensional complex vector subspaces of W ≅ ℂ2n, n ⩾ 3, and when X0 ⊂ X is the La grangian Grassmannian consisting of Lagrangian vector subspaces of (W, σ) where σ is an arbitrary symplectic form on W.

Local existence of a fourth-order dispersive curve flow on locally Hermitian symmetric spaces and its application

A fourth-order nonlinear dispersive partial differential equation arises in the field of mathematical physics, the solution of which is a curve flow on the two-dimensional unit sphere. In recent ten years, a geometric generalization of the sphere-valued physical model has been considered and the solvability of the initial value problem has been investigated. In particular, in the author's previous work, time-local existence and uniqueness result of the solution was established under the assumption that the solution is a closed curve flow on a compact Riemann surface with constant curvature. In the present paper, we propose a new geometric generalization of the sphere-valued physical model. As a main result, we show time-local existence of a solution to the initial value problem under the assumption that the solution is a closed curve flow on a compact locally Hermitian symmetric space. The proof is based on the geometric energy method combined with a gauge transformation to overcome the difficulty of the so-called loss of derivatives. Interestingly, the results can be applied to construct a generalized bi-Schrödinger flow proposed by Ding and Wang. The assumption on the manifold plays a crucial role both to enjoy a good solvable structure of the initial value problem and to reduce the generalized bi-Schrödinger flow equation to the one considered in the present paper.

Higher-order solutions of a matrix AKNS system associated with a Hermitian symmetric space

In this paper, we study a matrix Ablowitz–Kaup–Newell–Segur (AKNS) system associated with a Hermitian symmetric space as a follow-up study of an earlier paper. A multi-fold generalized Darboux transformation of the matrix AKNS system associated with a Hermitian symmetric space is constructed by means of determinants. Subsequently, we derive various higher-order solutions for this system, including fan-shaped rogue wave and (truncated) Kuznetsov–Ma breather solutions. Specifically, we show the fusion and fission processes for two truncated Kuznetsov–Ma breathers by taking different free parameters.

Hasimoto variables, generalized vortex filament equations, Heisenberg models and Schrödinger maps arising from group-invariant NLS systems

The deep geometrical relationships holding among the NLS equation, the vortex filament equation, the Heisenberg spin model, and the Schrödinger map equation are extended to the general setting of Hermitian symmetric spaces. New results are obtained by utilizing a generalized Hasimoto variable which arises from applying the general theory of parallel moving frames. The example of complex projective space ℂPN=SU(N+1)∕U(N) is used to illustrate the method and results.

Modulation instability and dynamics for the Hermitian symmetric space derivative nonlinear Schrödinger equation

In this paper, we analyse modulation instability of the Hermitian symmetric space derivative nonlinear Schrödinger equation. Based on the gauge transformation between the Lax pairs, we derive a classical and a generalized Darboux transformations for the Hermitian symmetric space derivative nonlinear Schrödinger equation and two compact determinant forms of the Darboux transformations by setting a restriction on spectral functions. Employing the two Darboux transformations, dynamical behaviors of all types of solutions of this equation are discussed in detail, such as single-hump soliton solutions, double-hump soliton solutions, eye-shaped rogue wave solutions, four-petaled rogue wave solutions, triangular rogue wave solutions and so on.