Showing papers on "Symplectic group published in 2013"

Higgs bundles and surface group representations in the real symplectic group

Abstract: In this paper, we study the moduli space of representations of a surface group (that is, the fundamental group of a closed oriented surface) in the real symplectic group Sp(2n, R). The moduli space is partitioned by an integer invariant, called the Toledo invariant. This invariant is bounded by a Milnor–Wood-type inequality. Our main result is a count of the number of connected components of the moduli space of maximal representations, that is, representations with maximal Toledo invariant. Our approach uses the non-abelian Hodge theory correspondence proved in a companion paper (O. Garc´oa-Prada, P. B. Gothen and I. Mundet i Riera, The Hitchin–Kobayashi correspondence, Higgs pairs and surface group representations, Preprint, 2012, arXiv:0909.4487 [math.AG].) to identify the space of representations with the moduli space of polystable Sp(2n, R)-Higgs bundles. A key step is provided by the discovery of new discrete invariants of maximal representations. These new invariants arise from an identification, in the maximal case, of the moduli space of Sp(2n, R)-Higgs bundles with a moduli space of twisted Higgs bundles for the group GL(n, R).

Metaplectic Anyons, Majorana Zero Modes, and their Computational Power

Abstract: We introduce and study a class of anyon models that are a natural generalization of Ising anyons and Majorana fermion zero modes. These models combine an Ising anyon sector with a sector associated with $SO(m)_2$ Chern-Simons theory. We show how they can arise in a simple scenario for electron fractionalization and give a complete account of their quasiparticles types, fusion rules, and braiding. We show that the image of the braid group is finite for a collection of $2n$ fundamental quasiparticles and is a proper subgroup of the metaplectic representation of $Sp(2n-2,\mathbb{F}_m)\ltimes H(2n-2,\mathbb{F}_m)$, where $Sp(2n-2,\mathbb{F}_m)$ is the symplectic group over the finite field $\mathbb{F}_m$ and $H(2n-2,\mathbb{F}_m)$ is the extra special group (also called the $(2n-1)$-dimensional Heisenberg group) over $\mathbb{F}_m$. Moreover, the braiding of fundamental quasiparticles can be efficiently simulated classically. However, computing the result of braiding a certain type of composite quasiparticle is $# P$-hard, although it is not universal for quantum computation because it has a finite braid group image. This a rare example of a topological phase that is not universal for quantum computation through braiding but nevertheless has $# P$-hard link invariants. We argue that our models are closely related to recent analyses finding non-Abelian anyonic properties for defects in quantum Hall systems, generalizing Majorana zero modes in quasi-1D systems.

Exponential Barycenters of the Canonical Cartan Connection and Invariant Means on Lie Groups

Abstract: When performing statistics on elements of sets that possess a particular geometric structure, it is desirable to respect this structure. For instance in a Lie group, it would be judicious to have a notion of a mean which is stable by the group operations (composition and inversion). Such a property is ensured for Riemannian center of mass in Lie groups endowed with a bi-invariant Riemannian metric, like compact Lie groups (e.g. rotations). However, bi-invariant Riemannian metrics do not exist for most non compact and non-commutative Lie groups. This is the case in particular for rigid-body transformations in any dimension greater than one, which form the most simple Lie group involved in biomedical image registration. In this chapter, we propose to replace the Riemannian metric by an affine connection structure on the group. We show that the canonical Cartan connections of a connected Lie group provides group geodesics which are completely consistent with the composition and inversion. With such a non-metric structure, the mean cannot be defined by minimizing the variance as in Riemannian Manifolds. However, the characterization of the mean as an exponential barycenter gives us an implicit definition of the mean using a general barycentric equation. Thanks to the properties of the canonical Cartan connection, this mean is naturally bi-invariant. We show the local existence and uniqueness of the invariant mean when the dispersion of the data is small enough. We also propose an iterative fixed point algorithm and demonstrate that the convergence to the invariant mean is at least linear. In the case of rigid-body transformations, we give a simple criterion for the global existence and uniqueness of the bi-invariant mean, which happens to be the same as for rotations. We also give closed forms for the bi-invariant mean in a number of simple but instructive cases, including 2D rigid transformations. For general linear transformations, we show that the bi-invariant mean is a generalization of the (scalar) geometric mean, since the determinant of the bi-invariant mean is the geometric mean of the determinants of the data. Finally, we extend the theory to higher order moments, in particular with the covariance which can be used to define a local bi-invariant Mahalanobis distance.

Uniqueness of symplectic structures

Abstract: This survey paper discusses some uniqueness questions for symplectic forms on compact manifolds without boundary.

The supersingular locus of the Shimura variety for GU(1,n-1) over a ramified prime

Abstract: We analyze the geometry of the supersingular locus of the reduction modulo p of a Shimura variety associated to a unitary similitude group GU(1,n-1) over Q, in the case that p is ramified. We define a stratification of this locus and show that its incidence complex is closely related to a certain Bruhat-Tits simplicial complex. Each stratum is isomorphic to a Deligne-Lusztig variety associated to some symplectic group over F_p and some Coxeter element. The closure of each stratum is a normal projective variety with at most isolated singularities. The results are analogous to those of Vollaard/Wedhorn in the case when p is inert.

On cohomological obstructions to the existence of log symplectic structures

Abstract: We prove that a compact log symplectic manifold has a class in the second cohomology group whose powers, except maybe for the top, are nontrivial This result gives cohomological obstructions for the existence of b-log symplectic structures similar to those in symplectic geometry

Grassmannian connection between three- and four-qubit observables, Mermin’s contextuality and black holes

Abstract: We invoke some ideas from finite geometry to map bijectively 135 heptads of mutually commuting three-qubit observables into 135 symmetric four -qubit ones. After labeling the elements of the former set in terms of a seven-dimensional Clifford algebra, we present the bijective map and most pronounced actions of the associated symplectic group on both sets in explicit forms. This formalism is then employed to shed novel light on recently-discovered structural and cardinality properties of an aggregate of three-qubit Mermin’s “magic” pentagrams. Moreover, some intriguing connections with the so-called black-hole-qubit correspondence are also pointed out.

Quantum Blobs

Abstract: Quantum blobs are the smallest phase space units of phase space compatible with the uncertainty principle of quantum mechanics and having the symplectic group as group of symmetries Quantum blobs are in a bijective correspondence with the squeezed coherent states from standard quantum mechanics, of which they are a phase space picture This allows us to propose a substitute for phase space in quantum mechanics We study the relationship between quantum blobs with a certain class of level sets defined by Fermi for the purpose of representing geometrically quantum states

The diversity of symplectic calabi-yau 6-manifolds

Abstract: Given an integer b and a finitely presented group G, we produce a compact symplectic 6-manifold with c1 = 0, b2 > b, b3 > b and pi = G. In the simply connected case, we can also arrange for b3 = 0; in particular, these examples are not diffeomorphic to Kahler manifolds with c1 = 0. The construction begins with a certain orientable, four-dimensional, hyperbolic orbifold assembled from right-angled 120-cells. The twistor space of the hyperbolic orbifold is a symplectic Calabi- Yau orbifold; a crepant resolution of this last orbifold produces a smooth symplectic manifold with the required properties. © 2013 London Mathematical Society.

Symplectic Bott-Chern cohomology of solvmanifolds

Abstract: We study the symplectic Bott-Chern cohomology by L.-S. Tseng and S.-T. Yau for solvmanifolds endowed with left-invariant symplectic structures. Our results are applicable to cohomology with values in local systems. Studying symplectic Bott-Chern cohomology of solvmanifolds with values in local systems, we give some remarks on symplectic Hodge theory.

Moments of Random Matrices and Weingarten Functions

Abstract: Let G be the unitary group, orthogonal group, or (compact) symplectic group, equipped with its Haar probability measure, and suppose that G is realized as a matrix group. Consider a random matrix X = (xi,j)1≤i,j≤N picked up from G. We would like to know how to compute the moments E [ xi1j1 · · ·xinjnxi′1j′ 1 · · ·xi′nj′ n ] or E [xi1j1 · · ·xi2nj2n ] . In this report, we focus on the unitary group UN and use the methods established in [5] and [9] which express the moments as sums in terms of Weingarten functions. The function Wg(·, N), called the unitary Weingarten function, has rich combinatorial structures involving Jucys-Murphy elements. We discuss and prove some of its properties. Finally, we consider some applications of the formula for integration with respect to the Haar measure over the unitary group UN . We compute matrix-valued expectations with the goal of having a better understanding of the operator-valued Cauchy transform.

Symplectic spreads, planar functions and mutually unbiased bases

Abstract: In this paper we give explicit descriptions of complete sets of mutually unbiased bases (MUBs) and orthogonal decompositions of special Lie algebras $sl_n(\mathbb{C})$ obtained from commutative and symplectic semifields, and from some other non-semifield symplectic spreads. Relations between various constructions are also studied. We show that the automorphism group of a complete set of MUBs is isomorphic to the automorphism group of the corresponding orthogonal decomposition of the Lie algebra $sl_n(\mathbb{C})$. In the case of symplectic spreads this automorphism group is determined by the automorphism group of the spread. By using the new notion of pseudo-planar functions over fields of characteristic two we give new explicit constructions of complete sets of MUBs.

Symplectic half-flat solvmanifolds

Abstract: We classify solvable Lie groups admitting left invariant symplectic half-flat structure. When the Lie group has a compact quotient by a lattice, we show that these structures provide solutions of supersymmetric equations of type IIA.

Grassmannian Connection Between Three- and Four-Qubit Observables, Mermin's Contextuality and Black Holes

Abstract: We invoke some ideas from finite geometry to map bijectively 135 heptads of mutually commuting three-qubit observables into 135 symmetric four-qubit ones. After labeling the elements of the former set in terms of a seven-dimensional Clifford algebra, we present the bijective map and most pronounced actions of the associated symplectic group on both sets in explicit forms. This formalism is then employed to shed novel light on recently-discovered structural and cardinality properties of an aggregate of three-qubit Mermin's 'magic' pentagrams. Moreover, some intriguing connections with the so-called black-hole--qubit correspondence are also pointed out.

Algorithms for data fitting on some common homogeneous spaces

Abstract: In signal processing, we have often to deal with data that belong to non-linear spaces, or manifolds. We can think about the rotation group in pose estimation, the diffusion tensors in image processing, or the Grassmann manifold in subspace tracking techniques. The goal of this thesis is to design algorithms to analyze this kind of data. We first consider averaging problems on Riemannian manifolds. In particular, we study gradient and Newton methods to compute the Karcher mean of rotation matrices and symmetric positive definite matrices. Then, we design a gradient method to compute a geodesic that best fits a set of time-labeled data points. We describe the approach on different symmetric spaces of interest in practical applications. Finally, on some Riemannian manifolds, there is no known algorithm to compute the Riemannian distance between two points. This is the case for the Stiefel manifold, the general linear group, or the symplectic group. We address these problems by developing some optimization schemes that enable us to compute this distance in some cases.

The homotopy Lie algebra of symplectomorphism groups of 3-fold blow-ups of the projective plane

Abstract: By a result of Kedra and Pinsonnault, we know that the topology of groups of symplectomorphisms of symplectic 4-manifolds is complicated in general. However, in all known (very specific) examples, the rational cohomology rings of symplectomorphism groups are finitely generated. In this paper, we compute the rational homotopy Lie algebra of symplectomorphism groups of the 3-point blow-up of the projective plane (with an arbitrary symplectic form) and show that in some cases, depending on the sizes of the blow-ups, it is infinite dimensional. Moreover, we explain how the topology is generated by the toric structures one can put on the manifold. Our method involve the study of the space of almost complex structures compatible with the symplectic structure and it depends on the inflation technique of Lalonde–McDuff.

The Maslov index in weak symplectic functional analysis

Abstract: We recall the Chernoff-Marsden definition of weak symplectic structure and give a rigorous treatment of the functional analysis and geometry of weak symplectic Banach spaces. We define the Maslov index of a continuous path of Fredholm pairs of Lagrangian subspaces in continuously varying Banach spaces. We derive basic properties of this Maslov index and emphasize the new features appearing.

Pieri algebras for the orthogonal and symplectic groups

Abstract: We study the structure of a family of algebras which encodes an iterated version of the Pieri Rule for the complex orthogonal group. In particular, we show that each of these algebras has a standard monomial basis and has a flat deformation to a Hibi algebra. There is also a parallel theory for the complex symplectic group.

Full automorphism group of the generalized symplectic graph

Abstract: Let F q be a finite field of odd characteristic, m, ν the integers with 1≤ m ≤ ν and K a 2 ν × 2 ν nonsingular alternate matrix over F q . In this paper, the generalized symplectic graph GS p 2 ν ( q,m ) relative to K over F q is introduced. It is the graph with m -dimensional totally isotropic subspaces of the 2 ν -dimensional symplectic space F q (2 ν ) as its vertices and two vertices P and Q are adjacent if and only if the rank of PKQ T is 1 and the dimension of P ∩ Q is m -1. It is proved that the full automorphism group of the graph GS p 2 ν ( q,m ) is the projective semilinear symplectic group PΣ p (2 ν, q ).

Non-Levi Closed Conjugacy Classes of SP q (2 n )

Abstract: We construct an explicit quantization of semi-simple conjugacy classes of the complex symplectic group SP(2n) with non-Levi isotropy subgroups through an operator realization on highest weight modules over the quantum group \({U_q\bigl(\mathfrak{sp}(2n)\bigr)}\).

On the structure of homogeneous symplectic varieties of complete intersection

Abstract: A normal complex algebraic variety X is called a symplectic variety if its regular locus Xreg admits a holomorphic symplectic 2-form ω such that it extends to a holomorphic 2-form on a resolution f : X → X . Affine symplectic varieties are constructed in various ways such as nilpotent orbit closures of a semisimple complex Lie algebra (cf. [CM]), Slodowy slices to nilpotent orbits (cf.[Sl]) or symplectic reductions of holomorphic symplectic manifolds with Hamiltonian actions. Usually these examples come up with C-actions. In this article we shall study a 2n-dimensional affine symplectic variety X ⊂ C defined as a complete intersection of r homogeneous polynomials fi(z1, ..., z2n+r) (1 ≤ i ≤ r). Here we assume that weights of all coordinates are 1: wt(z1) = ... = wt(z2n+r) = 1. The C -action on C induces a C-action on X . We also assume that the symplectic form ω is homogeneous with respect to thisC-action. Namely, for some integer l, we have tω = t ·ω where t ∈ C. The integer l is called the weight of ω and is denoted by wt(ω). A main result (Theorem 2) is that such an X is isomorphic (as a Cvariety) to a nilpotent orbit closure Ō of a semisimple complex Lie algebra and ω corresponds to the Kostant-Kirillov form on O. A nilpotent orbit closure does not generally have complete intersection singularities and the orbit that appears in Theorem 2 should be very restricted one. We conjecture that such a nilpotent orbit closure would be the nilpotent variety of a semisimple complex Lie algebra. As a particular case we consider a symplectic hypersurface, i.e. r = 1. As is studied in [LNSV] we have very few examples of quasihomogeneous symplectic hypersurfaces. In our homogeneous case it exists only when n = 1 and X must be isomorphic to an A1-surface singularity (Theorem 3).

Symplectic Lie Groups I-III

Abstract: We develop the structure theory of symplectic Lie groups based on the study of their isotropic normal subgroups. The article consists of three main parts. In the first part we show that every symplectic Lie group admits a sequence of subsequent symplectic reductions to a unique irreducible symplectic Lie group. The second part concerns the symplectic geometry of cotangent symplectic Lie groups and the theory of Lagrangian extensions of flat Lie groups. In the third part of the article we analyze the existence problem for Lagrangian normal subgroups in nilpotent symplectic Lie groups.

On the geometric side of the Arthur trace formula for the symplectic group of rank 2

Abstract: We study the non-semisimple terms in the geometric side of the Arthur trace formula for the split symplectic similitude group or the split symplectic group of rank 2 over any algebraic number field. In particular, we show that the coefficients of unipotent orbital integrals are expressed by the Dedekind zeta function, Hecke L-functions, and the Shintani zeta function for the space of binary quadratic forms.

An upper bound for the Lusternik-Schnirelmann category of the symplectic group

Abstract: We prove that the LS category of the symplectic group Sp(n) is bounded above by (n+1 \choose 2). This is achieved by computing the number of critical levels of a height function. © 2013 Cambridge Philosophical Society.

Classicité en théorie de Hida.

Abstract: In this article, we prove an effective classicity theorem in Hida's theory. We focus more precisely on the case of the symplectic group and Siegel modular forms. We also prove that certain mod~$p$ Siegel modular forms lift to characteristic zero and study carefully the divisibility by~$p$ of certain Hecke operators.

The Bruhat−Chevalley Order on Involutions of the Hyperoctahedral Group and the Combinatorics of B-Orbit Closures

Abstract: Let G = Sp2n (ℂ) be the symplectic group, B be its Borel subgroup, and Φ = C n be the root system of G. To each involution σ in the Weyl group W of Φ, one can assign an orbit Ω σ of the coadjoint action of B on the dual space of the Lie algebra of the unipotent radical of B. Let σ, τ be involutions in W. It is proved that Ω σ is contained in the closure of Ω τ if and only if σ is less than or equal to τ with respect to the Bruhat–Chevalley order on W. Bibliography: 15 titles.

Maximal Covariance Group of Wigner Transforms and Pseudo-Differential Operators

Abstract: We show that the linear symplectic and anti-symplectic transformations form the maximal covariance group for both the Wigner transform and Weyl operators. The proof is based on a new result from symplectic geometry which characterizes symplectic and anti-symplectic matrices, and which allows us, in addition, to refine a classical result on the preservation of symplectic capacities of ellipsoids.