Symplectic Case Research Articles

Strong asymptotic freeness for Wigner and Wishart matrices

For each n in N, let X n = [(X n ) jk ] n j,k=1 be a random Hermitian matrix such that the n 2 random variables √n(X n ) ii , √2nRe((X n ) ij ) i<j , √2n Im((X n ) ij ) i<j are independent, identically distributed, with common distribution μ on R. Let X (1) n ,..., x (r) n be r independent copies of X n and (x 1 ,...,x r ) be a semicircular system in a C*-probability space with a faithful state. Assuming that p is symmetric and satisfies a Poincare inequality, we show that, almost everywhere, for any non commutative polynomial p in r variables, (0.1) lim n→∞ ∥p(X (1) n ,..., X (r) n )∥ = ∥p(x 1 ,..., x r )∥. We follow the method of [10] and [17] which gave (0.1) in the Gaussian (complex, real or symplectic) case. We also get that (0.1) remains true when the X (i) n are Wishart matrices while the x i are Marchenko-Pastur distributed.

Non-commutative polynomials of independent Gaussian random matrices. The real and symplectic cases.

In [HT2] Haagerup and Thorbjo rnsen prove the following extension of Voiculescu’s random matrix model (cf. [V2, Theorem 2.2]): For each n ∈ ℕ, let X1(n),..., Xr(n) be r independent complex self-adjoint random matrices from the class Open image in new window and let x1,...,xr be a semicircular system in a C*-probability space. Then for any polynomial p in r non-commuting variables the convergence

Fermionic anticommutators for open superstrings in the presence of antisymmetric tensor field

We build up the anticommutator algebra for the fermionic coordinates of open superstrings attached to branes with antisymmetric tensor fields. We use both Dirac quantization and the symplectic Faddeev–Jackiw approach. In the symplectic case we find a way of generating the boundary conditions as zero modes of the symplectic matrix by taking a discretized form of the action and adding terms that vanish in the continuous limit. This way boundary conditions can be handled as constraints.

Constructing the symplectic Evans matrix using maximally analytic individual vectors

For linear systems with a multi-symplectic structure, arising from the linearization of Hamiltonian partial differential equations about a solitary wave, the Evans function can be characterized as the determinant of a matrix, and each entry of this matrix is a restricted symplectic form. This variant of the Evans function is useful for a geometric analysis of the linear stability problem. But, in general, this matrix of two-forms may have branch points at isolated points, shrinking the natural region of analyticity. In this paper, a new construction of the symplectic Evans matrix is presented, which is based on individual vectors but is analytic at the branch points—indeed, maximally analytic. In fact, this result has greater generality than just the symplectic case; it solves the following open problem in the literature: can the Evans function be constructed in a maximally analytic way when individual vectors are used? Although the non-symplectic case will be discussed in passing, the paper will concentrate on the symplectic case, where there are geometric reasons for evaluating the Evans function on individual vectors. This result simplifies and generalizes the multi-symplectic framework for the stability analysis of solitary waves, and some of the implications are discussed.

A symplectic case of Artin’s Conjecture

An equivalent form of this conjecture is the following. If ρ is irreducible and non-trivial, then L(s, ρ) is entire. Artin proved this conjecture in the case where ρ is 1-dimensional using his reciprocity law together with a result of Hecke. As L-functions are inductive, this proved the conjecture if ρ is monomial, i.e. induced from a degree one representation of some subgroup. Now consider the case where ρ is 2-dimensional. Let ρ : G → PGL2(C) be the composition of ρ : G → GL2(C) with the natural projection of GL2(C) to PGL2(C). Let Ḡ denote the image of G under ρ. So Ḡ is a finite subgroup of PGL2(C), which is isomorphic to SO3(C). The only finite subgroups of SO3(C) are cyclic, dihedral, tetrahedral (isomorphic to A4), octahedral (isomorphic to S4) and icosahedral (isomorphic to A5). We thus classify ρ according the isomorphism type of Ḡ. If Ḡ is cyclic or dihedral, then ρ is reducible or monomial and L(s, ρ) is entire by Artin’s result. Much later, Langlands applied to this problem the theory of automorphic representations, which also have associated L-functions. For cuspidal automorphic representations of GLn the associated L-function is known to be entire [Ja]. Both automorphic and Artin L-functions can be written as Euler products of local factors L(s, π) = ∏ v L(s, πv) and L(s, ρ) = ∏ v L(s, ρv). Langlands formulated the following amazing conjecture [La1].

Nearby Cycles for Local Models of Some Shimura Varieties

Kottwitz conjectured a formula for the (semi-simple) trace of Frobenius on the nearby cycles for the local model of a Shimura variety with Iwahori-type level structure. In this paper, we prove his conjecture in the linear and symplectic cases by adapting an argument of Gaitsgory, who proved an analogous theorem in the equal characteristic case.

Lefschetz type pencils on contact manifolds

We define the concept of Lefschetz contact pencil and we show the existence of such structures on any contact manifold. The main idea of the proof is a generalization of the Donaldson arguments used in the symplectic case. We will analyze some of the applications of such existence theorem for the topology of approximately holomorphic contact submanifolds.

RTT presentation of ortho-symplectic super Yangians. Staggered integrable spin chains)

We give a RTT presentation of ortho-symplectic super Yangians that encompasses the orthogonal and symplectic cases. In a second part, we construct an integrable ladder chain model and study its quantum symmetry.

Energy-level correlations in chiral symmetric disordered systems: Corrections to the universal results

We investigate the deviation of the level-correlation functions from the universal form for the chiral symmetric classes. Using the supersymmetric nonlinear \ensuremath{\sigma} model we formulate the perturbation theory. The large energy behavior is compared with the result of the diagrammatic perturbation theory. We have the diffuson and cooperon contributions even in the average density of states. For the unitary and orthogonal classes we get the small energy behavior that suggests a weakening of the level repulsion. For the symplectic case we get a result with opposite tendency.

ON THE COMPUTATION OF LYAPUNOV EXPONENTS FOR DISCRETE TIME SERIES: APPLICATIONS TO TWO-DIMENSIONAL SYMPLECTIC AND DISSIPATIVE MAPPINGS

Many techniques have been developed for the measure of the largest Lyapunov exponent of experimental short data series. The main idea, underlying the most common algorithms, is to mimic the method of computation proposed by Benettin and Galgani [1979]. The aim of the present paper is to provide an explanation for the reliability of some algorithms developed for short time series. To this end, we consider two-dimensional mappings as model problems and we compare the results obtained using the Benettin and Galgani method to those obtained using some algorithms for the computation of the largest Lyapunov exponent when dealing with short data series. In particular we focus our attention on conservative systems, which are not widely investigated in the literature. We show that using standard techniques the results obtained for discrete series related to area-preserving mappings are often unreliable, while dissipative systems are easier to analyze. In order to overcome the problem arising with conservative systems, we develop an alternative method, which takes advantage of the existing techniques. In particular, all algorithms provide a good approximation of the largest Lyapunov exponent in the strong chaotic symplectic case and in the dissipative one. However, the application of standard algorithms provides results which are not in agreement with the theoretical expectation for weak chaotic motions, and sometimes also for regular orbits. On the contrary, the method that we propose in this paper seems to work well for the weak chaotic case. Because of the speed of computation, we suggest to use all algorithms to cross-check the results.

Fake symmetry transitions in lattice Dirac spectra

In a recent lattice investigation of Ginsparg-Wilson-type Dirac operators in the Schwinger model, it was found that the symmetry class of the random matrix theory describing the small Dirac eigenvalues appeared to change from the unitary to the symplectic case as a function of lattice size and coupling constant. We present a natural explanation for this observation in the framework of a random matrix model, showing that the apparent change is caused by the onset of chiral symmetry restoration in a finite volume. A transition from unitary to symplectic symmetry does not occur.

Exact Wigner Surmise Type Evaluation of the Spacing Distribution in the Bulk of the Scaled Random Matrix Ensembles

Random matrix ensembles with orthogonal and unitary symmetry correspond to the cases of real symmetric and Hermitian random matrices, respectively. We show that the probability density function for the corresponding spacings between consecutive eigenvalues can be written exactly in the Wigner surmise type form a(s)e−b(s) for a simply related to a Painleve transcendent and b its anti-derivative. A formula consisting of the sum of two such terms is given for the symplectic case (Hermitian matrices with real quaternion elements).

Level statistics inside the vortex of a superconductor and symplectic random-matrix theory in an external source

In the core of the vortex of a superconductor, energy levels appear inside the gap. We discuss here through a random matrix approach how these levels are broadened by impurities. It is first shown that the level statistics is governed by an ensemble consisting of a symplectic random potential added to a non-random matrix. A generalization of previous work on the unitary ensemble in the presence of an external source (which relied on the Itzykson-Zuber integral) is discussed for this symplectic case through the formalism introduced by Harish-Chandra and Duistermaat-Heckman. This leads to explicit formulae for the density of states and for the correlation functions, which describe the cross-over from the clean to the dirty limits.

On the Group of Diffeomorphisms Preserving a Locally Conformal Symplectic Structure

The automorphism group of a locally conformal symplectic structure is studied. It is shown that this group possesses essential features of the symplectomorphism group. By using a special type of cohomology the flux and Calabi homomorphisms are introduced. The main theorem states that the kernels of these homomorphisms are simple groups (for the precise statement, see Section 7). Some of the methods used may also be interesting in the symplectic case.

Poisson structures on Hilbert schemes of points¶of a surface and integrable systems

In this paper we prove that if S is a Poisson surface, i.e., a smooth algebraic surface with a Poisson structure, the Hilbert scheme of points of S has a natural Poisson structure, induced by the one of S. This generalizes previous results obtained by A. Beauville [B1] and S. Mukai [M2] in the symplectic case, i.e., when S is an abelian or K3 surface. Finally we apply our results to give some examples of integrable Hamiltonian systems naturally defined on these Hilbert schemes. In the simple case S=ℙ2 we obtain by this construction a large class of integrable systems, which includes the ones studied by P. Vanhaecke in [V1] and, more generally, in [V2].

Weakly symmetric spaces and bounded symmetric domains

LetG be a connected, simply-connected, real semisimple Lie group andK a maximal compactly embedded subgroup ofG such thatD=G/K is a hermitian symmetric space. Consider the principal fiber bundleM=G/K s →G/K, whereK s is the semisimple part ofK=K s ·Z K 0 andZ K 0 is the connected center ofK. The natural action ofG onM extends to an action ofG 1=G×Z K 0 . We prove as the main result thatM is weakly symmetric with respect toG 1 and complex conjugation. In the case whereD is an irreducible classical bounded symmetric domain andG is a classical matrix Lie group under a suitable quotient, we provide an explicit construction ofM=D×S 1 and determine a one-parameter family of Riemannian metrics Ω onM invariant underG 1. Furthermore,M is irreducible with respect to Ω. As a result, this provides new examples of weakly symmetric spaces that are nonsymmetric, including those already discovered by Selberg (cf. [M]) for the symplectic case and Berndt and Vanhecke [BV1] for the rank-one case.

Quantum relaxation in open chaotic systems

Using the supersymmetry technique, we analytically derive the recent result of Casati, Maspero, and Shepelyansky [Phys. Rev. E 56, R6233 (1997)] according to which the quantum dynamics of open chaotic systems follows the classical decay up to a new quantum relaxation time scale ${t}_{q}\ensuremath{\sim}\sqrt{{t}_{c}{t}_{H}}.$ This scale is larger than the classical escape time ${t}_{c}$ but still much smaller than the Heisenberg time ${t}_{H}.$ For systems with orthogonal or unitary symmetry the quantum decay is slower than the classical one while for the symplectic case there is an intermediate regime in which the quantum decay is slightly faster.

Sub-weakly Embedded Singular and Degenerate Polar Spaces

We show that every sub-weak embedding of any singular (degenerate or not) orthogonal or unitary polar space of non-singular rank at least 3 in a projective space PG $$(d,\mathbb{K}) $$ , $$\mathbb{K} $$ a commutative field, is the projection of a full embedding in some subspace PG $$\left( \overline {d}, \mathbb{F} \right) $$ of PG $$\left( \overline {d}, \mathbb{K} \right) $$ , where PG $$\left( \overline {d}, \mathbb{K} \right) $$ contains PG $$(d,\mathbb{K}) $$ and $$\mathbb{F} $$ is a subfield of $$\mathbb{K} $$ . The same result is proved in the symplectic case under the assumption that the field over which the polarity is defined is perfect if the characteristic is 2 and if each secant line of the embedded polar space Γ contains exactly two points of Γ. This completes the classification of all sub-weak embeddings of orthogonal, symplectic and unitary polar spaces (singular or not; degenerate or not) of non-singular rank at least 3 and defined over a commutative field $$\mathbb{F}\prime $$ , where in the characteristic 2 case $$\mathbb{F}\prime $$ is perfect if the polar space is symplectic and the degree of the embedding is 2.

Kählerian potentials and convexity properties of the moment map

Since Kostant proved his convexity theorems for torus actions on ag varieties ([Ko]), there have been numerous contributions to this subject in the more general symplectic and K ahlerian setting. For example, let K be a compact Lie group acting in a Hamiltonian fashion on a connected compact symplectic manifold X . Then the intersection (X )+ of the image of the moment map : X → (LieK)∗ with a positive Weyl chamber t∗ + is convex. Thus (X ) = K · (X )+, where (X )+ is a natural convex section. This was proved by Atiyah and Guillemin–Sternberg ([A], [G-S]) in the abelian case, i.e., where K = T is a compact torus, by Mumford [M]) for X projective algebraic with an integral K ahlerian structure and K not necessarily abelian and in its nal form in the compact symplectic case by Kirwan ([K]). A precise description of (X )+ := (X ) ∩ t∗ + in the compact K ahlerian framework can be given in terms of the image of the components of the set of T xed points in X . In the projective case, where the K ahlerian structure comes from a projective embedding, there are explicit connections to the representations theory of G = KC which are due to Brion and to Mumford–Ness ([B], [N]). The purpose of the present paper is to prove convexity properties of in the setting of non-compact K ahlerian spaces. Note that, by removing an appropriate K-invariant subset from X one can easily construct non-convex -images from convex ones. On the other hand Hilgert–Neeb–Plank proved ([H-N-P]) that convexity holds for a proper moment map : X → (LieK)∗. Another convexity result has been obtained by Sjamaar for X a ne and the restriction of the moment map of a representation ([S2]). Here the representation theoretical approach of Brion is used. Now, let X be an irreducible complex space endowed with a holomorphic action of a complex reductive group G. Assume that a maximal

Critical exponent of the localization length for the symplectic case

A new summability method was tested to calculate the critical exponent of the localization length for the symplectic case derived from the nonlinear -model. Although we used the same series as Hikami and others, unlike them we were able to resum the series in two dimensions (2D) and obtain the result . Values of in dimensions seem to saturate the Harris inequality up to .