Abelian Representations Research Articles

On representations of [formula omitted], [formula omitted] and [formula omitted

By work of Belyĭ [2], the absolute Galois group GQ=Gal(Q‾/Q) of the field Q of rational numbers can be embedded into A=Aut(Fˆ2), the automorphism group of the free profinite group Fˆ2 on two generators. The image of GQ lies inside GTˆ, the Grothendieck-Teichmüller group. While it is known that every abelian representation of GQ can be extended to GTˆ, Lochak and Schneps [13] put forward the challenge of constructing irreducible non-abelian representations of GTˆ. We do this virtually, namely by showing that a rich class of arithmetically defined representations of GQ can be extended to finite index subgroups of GTˆ. This is achieved, in fact, by extending these representations all the way to finite index subgroups of A=Aut(Fˆ2). We do this by developing a profinite version of the work of Grunewald and Lubotzky [7], which provided a rich collection of representations for the discrete group Aut(Fd).

Abelian Fock - Schwinger representation for the quark propagator in external gauge field

We develop the Diakonov - Petrov approach to the representation of the non - Abelian Wilson loop in an Abelian form. First of all, we extend this approach to the Abelian representation of the parallel transporter corresponding to the open curve rather than to the closed contour. Next, we extend this representation to the case of the non - compact groups. The obtained expressions allow us to derive the Abelian form of the Schwinger - Fock representation for the quark propagator in an external color gauge field.

On high-dimensional representations of knot groups

Given a hyperbolic knot $K$ and any $n\geq 2$ the abelian representations and the holonomy representation each give rise to an $(n-1)$-dimensional component in the $\operatorname{SL}(n,\Bbb{C})$-character variety. A component of the $\operatorname{SL}(n,\Bbb{C})$-character variety of dimension $\geq n$ is called high-dimensional. It was proved by Cooper and Long that there exist hyperbolic knots with high-dimensional components in the $\operatorname{SL}(2,\Bbb{C})$-character variety. We show that given any non-trivial knot $K$ and sufficiently large $n$ the $\operatorname{SL}(n,\Bbb{C})$-character variety of $K$ admits high-dimensional components.

On the traceless SU(2) character variety of the 6-punctured 2-sphere

We exhibit the traceless SU(2) character variety of a 6-punctured 2-sphere as a 2-fold branched cover of [Formula: see text], branched over the singular Kummer surface, with the branch locus in [Formula: see text] corresponding to the binary dihedral representations. This follows from an analysis of the map induced on SU(2) character varieties by the 2-fold branched cover [Formula: see text] branched over [Formula: see text] points, combined with the theorem of Narasimhan–Ramanan which identifies [Formula: see text] with [Formula: see text]. The singular points of [Formula: see text] correspond to abelian representations, and we prove that each has a neighborhood in [Formula: see text] homeomorphic to a cone on [Formula: see text].

Knot state asymptotics II: Witten conjecture and irreducible representations

This article pursues the study of the knot state asymptotics in the large level limit initiated in Charles and Marché (Knot state asymptotics I. Abelian representations and the A–J conjecture, 2011). As a main result, we prove the Witten asymptotic expansion conjecture for the Dehn fillings of the figure eight knot. The state of a knot is defined in the realm of Chern-Simons topological quantum field theory as a holomorphic section on the $\operatorname {SU}_{2}$ -character manifold of the peripheral torus. In the previous paper, we conjectured that the knot state concentrates on the character variety of the knot with a given asymptotic behavior on the neighborhood of the abelian representations. In the present paper we study the neighborhood of irreducible representations. We conjecture that the knot state is Lagrangian with a phase and a symbol given respectively by the Chern-Simons and Reidemeister torsion invariants. We show that under some mild assumptions, these conjectures imply the Witten conjecture on the asymptotic expansion of WRT invariants of the Dehn fillings of the knot. Using microlocal techniques, we show that the figure eight knot state satisfies our conjecture starting from q-differential relations verified by the colored Jones polynomials. The proof relies on a differential equation satisfied by the Reidemeister torsion along the branches of the character variety, a phenomenon which has not been observed previously as far as we know.

Knot state asymptotics I: AJ conjecture and Abelian representations

Consider the Chern-Simons topological quantum field theory with gauge group $\operatorname {SU}_{2}$ and level k. Given a knot in the 3-sphere, this theory associates to the knot exterior an element in a vector space. We call this vector the knot state and study its asymptotic properties when the level is large. The latter vector space being isomorphic to the geometric quantization of the $\operatorname {SU}_{2}$ -character variety of the peripheral torus, the knot state may be viewed as a section defined over this character variety. We first conjecture that the knot state concentrates in the large level limit to the character variety of the knot. This statement may be viewed as a real and smooth version of the AJ conjecture. Our second conjecture says that the knot state in the neighborhood of Abelian representations is a Lagrangian state. Using microlocal techniques, we prove these conjectures for the figure eight and torus knots. The proof is based on q-difference relations for the colored Jones polynomial. We also provide a new proof for the asymptotics of the Witten-Reshetikhin-Turaev invariant of the lens spaces and a derivation of the Melvin-Morton-Rozansky theorem from the two conjectures.

Traces, Dispersions of States and Hidden Variables

The interplay between the tracial property and minimality of dispersions of states on projections of von Neumann algebras and C*-algebras is investigated. Let φ be a state on a C*-algebra A with the projection structure P(A). The dispersion σ(φ) is defined as σ(φ) = sup{φ(p) − φ(p)2 | p ɛ P(A)}. It is proved that σ(φ) ≥ 2/9 whenever φ is a state on a real rank zero C*-algebra with no nonzero abelian representation. New characterization of traces in terms of dispersions is proved: A state on a von Neumann algebra without abelian and Type I2 direct summands is a trace if and only if φ has the minimal dispersion on all 3x3 matrix substructures. A similar characterization of semifinite normal traces on von Neumann algebras is obtained. The connection between unitary invariance of states and minimal dispersion property on C*-algebras is studied. Besides providing a new characterization of trace in terms of physically relevant properties, the existing results on hidden variables in W*- and C*-formalism of quantum mechanics are strengthen.

Abelian representation for the non-Abelian Wilson loop and the non-Abelian Stokes theorem on the lattice

We derive the Abelian - like expression for the lattice SU(N) Wilson loop in arbitrary irreducible representation. The continuum Abelian representation of the SU(N) Wilson loop (for the loop without selfintersections) that has been obtained by Diakonov and Petrov appears to be a continuum limit of this expression. We also obtain the lattice variant of a non - Abelian Stokes theorem and present the explicit expression for the matrix $\cal H$ used in the Diakonov - Petrov approach.

Varieties of representations of virtual knot groups in [formula omitted

We study the local structure of the variety of representations of a virtual knot group in SL2(C) near an abelian representation ρ0. To such a representation is attached a complex number ω and there are three cases. If ω and ω−1 are not roots of the Alexander polynomial, there are only abelian representations around ρ0. If ω is a root and ω−1 is not, there are only reducible representations. If both ω and ω−1 are roots and certain homological conditions hold, there are irreducible representations.

INVOLUTIONS ON KNOT GROUPS AND VARIETIES OF REPRESENTATIONS IN A LIE GROUP

We prove the existence of a rationalisation [Formula: see text] of a classical or high-dimensional knot group Π which admits an involution if the Alexander polynomials of the knot are reciprocal. Using the group [Formula: see text] and its involution, we study the local structure, in the neighbourhood of an abelian representation, of the space of representation of the knot group Π in a a Lie group. We apply these results to the groups of classical prime knots up to 10 crossings.

Arcs de représentations du groupe d'un nœud dans un groupe de Lie

We give sufficient conditions for the existence of an arc of non-metabelian representations with endpoint a given abelian representation of a knot group into a complex connected reductive Lie group. We determine also the quadratic and tangent cones at this representation and show that they coincide in certain cases.

Deforming abelian SU(2)-representations of knot groups

The aim of this paper is to generalize at heorem of C. D. Frohman and E. P. Klassen ((FK91)) concerning deformations of abelian SU(2)-representations of knot groups into non- abelian representations. The proof of our main theorem makes use of a generalization of a result of X.-S. Lin ((Lin92)) which should be interesting in itself.

BRS cohomology in string theory: Geometry of Abelization and the quartet mechanism

The Abelian representation of the string Becchi–Rouet–Stora (BRS) cohomology in terms of basic operators naturally forming two dual Kugo–Ojima quartets is found. Furthermore, using a gauge-fixing scheme, a contracting homotopy operator associated with the string BRS operator is constructed. This formalism gives the explicit realizations of the physical and unphysical subspaces. The passage between Abelian and non-Abelian quantities is realized geometrically through the use of the moments of the vertex operator, which act as vielbeins between infinite-dimensional algebras. The connection between the Virasoro and spectrum-generating algebras is clarified and the algebraic duality relation between them is uncovered.

Kernels of Free Abelian Representations of a Link Group

Kernels of Free Abelian Representations of a Link Group

Kernels of free abelian representations of a link group

foreachi=1, 2, * * ,u It is the purpose of this paper to determine the structure of the kernels of these homomorphisms. In order to do this, we will construct infinite cyclic covering spaces over S3-L using methods similar to those of Lee Neuwirth in [2]. Using these infinite cyclic covering spaces, we will observe how to construct covering spaces over S3 L corresponding to the kernels of homomorphisms G->Ail 0 ... (A m, 1_ ij< y, and, under certain conditions, it will be possible to determine the structure of these kernels, in particular, of the commutator subgroup of G.