Hermitian Symmetric Spaces Research Articles

Ricci-flat Kähler manifolds from supersymmetric gauge theories

Using techniques of supersymmetric gauge theories, we present the Ricci-flat metrics on non-compact Kähler manifolds whose conical singularity is repaired by the Hermitian symmetric space. These manifolds can be identified as the complex line bundles over the Hermitian symmetric spaces. Each of the metrics contains a resolution parameter which controls the size of these base manifolds, and the conical singularity appears when the parameter vanishes.

The Maslov index revisited

LetD be a Hermitian symmetric space of tube type,S its Shilov boundary andG the neutral component of the group of bi-holomorphic diffeomorphisms ofD. In the model situationD is the Siegel disc,S is the manifold of Lagrangian subspaces andG is the symplectic group. We introduce a notion of transversality for pairs of elements inS, and then study the action ofG on the set of triples of mutually transversal points inS. We show that there is a finite number ofG-orbits, and to each orbit we associate an integer, thus generalizing theMaslov index. Using the scalar automorphy kernel ofD, we construct a ℂ*,G-invariant kernel onD×D×D. Taking a specific determination of its argument and studying its limit when approaching the Shilov boundary, we are able to define a ℤ-valued,G-invariant kernel for triples of mutually transversal points inS. It is shown to coincide with the Maslov index. Symmetry properties and cocycle properties of the Maslov index are then easily obtained.

On the reductive Borel-Serre compactification: L p -cohomology of arithmetic groups (for large p )

The L 2 -cohomology of an arithmetic quotient of an Hermitian symmetric space (i.e., of a locally symmetric variety) is known to have the topological interpretation as the intersection homology of its Baily-Borel Satake compactification. In this article, we observe that even without the Hermitian hypothesis, the L p -cohomology of an arithmetic quotient, for p finite and sufficiently large, is isomorphic to the ordinary cohomology of its reductive Borel-Serre compactification. We use this to generalize a theorem of Mumford concerning homogeneous vector bundles, their invariant Chern forms and the canonical extensions of the bundles; here, though, we are referring to canonical extensions to the reductive Borel-Serre compactification of any arithmetic quotient. To achieve that, we give a systematic discussion of vector bundles and Chern classes on stratified spaces.

Darboux transformation and Crum’s formula for multi-component integrable equations

We present the Darboux transformation and the N-soliton solution for the multi-component integrable equations which are associated with the Hermitian symmetric spaces. Using the Darboux covariance, we derive new integrable equations as well as known ones including the multi-component extensions of the nonlinear Schrödinger, the modified KdV, the SIT and the sine-Gordon equations. We also derive a closed form of the N-soliton solution in terms of the generalized Crum’s formula. The projection property of the Darboux transformation is explained.

Un théorème de rigidité non-métrique pour les variétés localement symétriques hermitiennes

Let X be an irreducible Hermitian symmetric space of non-compact type of dimension greater than 1 and G be the group of biholomorphisms of X ; let {\rm M} = \Gamma \backslash X be a quotient of X by a torsion-free discrete subgroup \Gamma of G such that M is of finite volume in the canonical metric. Then, due to the G-equivariant Borel embedding of X into its compact dual Xc, the locally symmetric structure of M can be considered as a special kind of a (G_{\Bbb C} , X_c) -structure on M, a maximal atlas of Xc-valued charts with locally constant transition maps in the complexified group {\rm G}_{\Bbb C} . By Mostow's rigidity theorem the locally symmetric structure of M is unique. We prove that the ({\rm G}_{\Bbb C} , X_c) -structure of M is the unique one compatible with its complex structure. In the rank one case this result is due to Mok and Yeung.

Generalized Heisenberg ferromagnet models via Hermitian symmetric spaces

We study the generalized Heisenberg ferromagnet model using a duality with the Hermitian symmetric space nonlinear Schrodinger equation. We explain the mathematical structure of the Hermiticity of the space in connection with the soliton surface theory and obtain various spin configurations in N-dimension using this correspondence.

Deformation of surfaces, integrable systems, and Chern–Simons theory

A few years ago, some of us devised a method to obtain integrable systems in (2+1)-dimensions from the classical non-Abelian pure Chern–Simons action via the reduction of the gauge connection in Hermitian symmetric spaces. In this article we show that the methods developed in studying classical non-Abelian pure Chern–Simons actions can be naturally implemented by means of a geometrical interpretation of such systems. The Chern–Simons equation of motion turns out to be related to time evolving two-dimensional surfaces in such a way that these deformations are both locally compatible with the Gauss–Mainardi–Codazzi equations and completely integrable. The properties of these relationships are investigated together with the most relevant consequences. Explicit examples of integrable surface deformations are displayed and discussed.

Shimura invariant differential operators and their eigenvalues

Let $\bar D$ be the covariant Cauchy-Riemann operator and $\mathcal D$ the covariant holomorphic differential operator on a line bundle over a Hermitian symmetric space $G/K$ . We study the Shimura invariant differential operators defined via $\bar D$ and $\mathcal D$ . We find the eigenvalues of a family of the Shimura operators and of the generators.

Large-N Limit of N=2 Supersymmetric QN Model in Two Dimensions

We investigate non-perturbative structures of the two-dimensional N=2 supersymmetric nonlinear sigma model on the quadric surface Q^{n-2}(C) = SO(n)/SO(n-2)xU(1), which is a Hermitian symmetric space, and therefore Kahler, by using the auxiliary field and large-n methods. This model contains two kinds of non-perturbatively stable vacua; one of them is the same vacuum as that of supersymmetric CP^{n-1} model, and the other is a new kind of vacuum, which has not yet been known to exist in two-dimensional nonlinear sigma models, the Higgs phase. We show that both of these vacua are asymptotically free. Although symmetries are broken in these vacua, there appear no massless Nambu-Goldstone bosons, in agreement with Coleman's theorem, due to the existence of two different mechanisms in these vacua, the Schwinger and the Higgs mechanisms.

Effective estimates on the very ampleness of the canonical line bundle of locally Hermitian symmetric spaces

We study the problem about the very ampleness of the canonical line bundle of compact locally Hermitian symmetric manifolds of non-compact type. In particular, we show that any suciently large unramied covering of such manifolds has very ample canonical line bundle, and give estimates on the size of the covering manifold, which is itself a locally Hermitian symmetric manifold, in terms of geometric data such as injectivity radius or degree of coverings. Let L be an ample line bundle on an algebraic manifold M. From Kodaira's Embedding Theorem, we know that mL is very ample if m is suciently large so that the sections of mL give rise to an embedding of M into some projective space. A natural question is on the estimates of such m so that mL is very ample. In particular, one may ask the same question for the canonical line bundle of a manifold with negative rst Chern class, or of general type. Of particular interest among such manifolds is the class of locally Hermitian symmetric spaces. The main purpose of this article is to show that for compact locally Hermitian symmetric manifolds of non-compact type, K is very ample if the injectivity radius of the manifold is greater than some eective constant which can be estimated in terms of some geometric data. Eective estimates for mK = K +( m 1)K with m 2 in terms of injectivity radius have been obtained in (HT) and (Y2). The diculty in the case of m =1c an be seen from the fact that in constructing holomorphic sections from the Bochner- Kodaira technique, one always needK +L for some extra positivity fromL .I n (Y2), it is shown that for a tower of coverings of compact Hermitian locally symmetric manifolds Mi, KMi becomes very ample when i approaches to1. To compensate for the extra positivity required for the Bochner-Kodaira technique, we prove our result by showing that the limit of the Bergman kernel on Mi approaches the corresponding one on the universal covering. However, the limiting process makes the whole approach highly non-eective and requires an innite sequence of normal coverings of the original manifold. The main purpose of this article is to show the very ampleness of KM for any unramied covering M of Mo with injectivity radius or degree of the coverings greater than some eectively estimable constant. For this purpose, an alternate approach using heat kernel estimates combined with Atiyah's

Invariant differential operators on Hermitian symmetric spaces and their eigenvalues

Let\(\bar D\) be the invariant Cauchy Riemann operator and\(\mathcal{M}_m = D^m \bar D^m \) the corresponding invariant Laplacians on a bounded symmetric domain. We calculate the eigenvalues ofM m on spherical functions. In particular we prove that for a symmetric domain of rank two the operatorsM 1,M 3 generate all invariant differential operators. We also find the eigenvalues of the generators introduced by Shimura.

Geometric realizations of Fordy–Kulish nonlinear Schrödinger systems

A method of Sym and Pohlmeyer, which produces geometric realizations of many integrable systems, is applied to the Fordy-Kulish generalized non-linear Schrodinger systems associated with Hermitian symmetric spaces. The resulting geometric equations correspond to distinguished arclength-parametrized curves evolving in a Lie algebra, generalizing the localized induction model of vortex filament motion. A natural Frenet theory for such curves is formulated, and the general correspondence between curve evolution and natural curvature evolution is analyzed by means of a geometric recursion operator. An appropriate specialization in the context of the symmetric space SO(p + 2)/SO(p) × SO(2) yields evolution equations for curves in Rp+1 and Sp, with natural curvatures satisfying a generalized mKdV system. This example is related to recent constructions of Doliwa and Santini and illuminates certain features of the latter. ∗Dept. of Mathematics, Case Western Reserve University, Cleveland OH 44106 email: jxl6@po.cwru.edu †Dept. of Mathematics and Comp. Sci., Drexel University, Philadelphia PA 19104 email: rperline@mcs.drexel.edu Mathematics Subject Classification: 58F07, 35Q55

Applications of Bruhat decompositions to complex hyperbolic geometry

The double coset space AΛ (n, ℂ) / U (n − 1, 1) is studied, where A consists of the diagonal matrices in GL (n, ℂ). This space naturally arises in the harmonic analysis on the hermitian symmetric space GL (n, ℂ) / U (n − 1, 1). It is shown here that these double cosets also represent a class of basic invariants related to complex hyperbolic geometry. An algebraic parametrization for the double cosets is given and it is shown how this may be used to conveniently compute the geometric invariants.

Hermitian-Einstein metrics for vector bundles on complete Kähler manifolds

In this paper, we prove the existence of Hermitian-Einstein metrics for holomorphic vector bundles on a class of complete Kähler manifolds which include Hermitian symmetric spaces of noncompact type without Euclidean factor, strictly pseudoconvex domains with Bergman metrics and the universal cover of Gromov hyperbolic manifolds etc. We also solve the Dirichlet problem at infinity for the Hermitian-Einstein equations on holomorphic vector bundles over strictly pseudoconvex domains.

Dimensionally reduced gravity,Hermitian symmetric spaces andtheAshtekar variables

Dimensional reductions of various higher-dimensional (super)gravitytheories lead toeffectively two-dimensional field theories described by gravity-coupledG/H nonlinear σ-models. We show that a new set ofcomplexifiedvariablescan be introduced when G/H is aHermitian symmetric space.This generalizes an earlier construction that grew out of the Ashtekarformulation of two-Killing-vector reduced pure four-dimensional generalrelativity.Apart from givingsome new insights into dimensional reductions of higher-dimensional(super)gravity theories, these Ashtekar-type variablesoffer several technical advantages in the context of the exactquantization of these models. As an application, an infiniteset of conserved charges is constructed. Our results might serve as astarting point for probing the quantum equivalence of theAshtekar and the metric formalism within a non-trivial midi-superspacemodel of quantum gravity.

Differential geometry of Grassmannian embeddings of based loop groups

In this article we study differential geometric properties of the most basic infinite-dimensional manifolds arising from fermionic (1+1) -dimensional quantum field theory: the restricted Grassmannian and the group of based loops in a compact simple Lie group. We determine the Riemann curvature tensor and the (linearly) divergent expression corresponding to the Ricci curvature of the restricted Grassmannian after proving that the latter manifold is an isotropy irreducible Hermitian symmetric space. Using the Gauss equation of the embedding of a based loop group into the restricted Grassmannian we show that the (conditional) Ricci curvature of a based loop group is proportional to its metric. Furthermore we explicitly derive the logarithmically divergent behaviour of several differential geometric quantities arising from this embedding.

Complete nondegenerate locally standard CR manifolds

Abstract. Let M be a complete nondegenerate locally standard CR manifold. We show that a necessary and sufficient condition for M to be compact is that the Lie algebra of its infinitesimal CR automorphisms is semisimple. In general we realize M as a Mostow fibration over a compact CR manifoldB whose universal covering is a Cartesian product of Hermitian symmetric spaces and compact nondegenerate standard CR manifolds.

Asphericity of Moduli Spaces via Curvature

We show that under suitable conditions a branched cover satisfies the same upper curvature bounds as its base space. First we do this when the base space is a metric space satisfying Alexandrov's curvature condition CAT(κ) and the branch locus is complete and convex. Then we treat branched covers of a Riemannian manifold over suitable mutually orthogonal submanifolds. In neither setting do we require that the branching be locally finite. We apply our results to hyperplane complements in several Hermitian symmetric spaces of nonpositive sectional curvature in order to prove that two moduli spaces arising in algebraic geometry are aspherical. These are the moduli spaces of the smooth cubic surfaces in ℂP3 and of the smooth complex Enriques surfaces.

Simpleness and closedness of circles in compact Hermitian symmetric spaces

We first interpret circles in Riemannian Symmetric space by Lie algebro-theoretic formalism. In particular, it is a solution of the system of ordinary differential equation of first order. We divide circles into 3-types. We investigate closedness and simpleness for such circles in compact Hermitian symmetric spaces. Consequently, we find many open holomorphic circles and non-simple circles. Note that there exist no non-simple circles and no open holomorphic circles in compact Riemannian symmetric space of rank one.

Supersymmetric Nonlinear Sigma Models as Gauge Theories

Supersymmetric nonlinear sigma models are obtained from linear sigma models by imposing supersymmetric constraints. If we introduce auxiliary chiral and vector superfields, these constraints can be expressed by D-terms and F-terms depending on the target manifolds. Auxiliary vector superfields appear as gauge fields without kinetic terms. If there are no D-term constraints, the target manifolds are always non-compact manifolds. When all the degrees of freedom in these non-compact directions are eliminated by gauge symmetries, the target manifold becomes compact. All supersymmetric nonlinear sigma models, whose target manifolds are the hermitian symmetric spaces, are successfully formulated as gauge theories.