6j Symbols for the Modular Double,Quantum Hyperbolic Geometry,and Supersymmetric Gauge Theories

We revisit the definition of the 6j symbols from the modular double of ${\mathcal{U}_q(\mathfrak{sl}(2, \mathbb{R}))}$ , referred to as b-6j symbols. Our new results are (1) the identification of particularly natural normalization conditions, and (2) new integral representations for this object. This is used to briefly discuss possible applications to quantum hyperbolic geometry, and to the study of certain supersymmetric gauge theories. We show, in particular, that the b-6j symbol has leading semiclassical asymptotics given by the volume of a non-ideal tetrahedron. Our new integral representations finally indicate a possible interpretation of the b-6j symbols as partition functions of non-abelian three-dimensional ${\mathcal{N}=2}$ supersymmetric gauge theories.     

Trivalent Graphs,Volume Conjectures and Character Varieties

The generalized volume conjecture and the AJ conjecture (a.k.a. the quantum volume conjecture) are extended to \({U_q(\mathfrak{sl}_2)}\) colored quantum invariants of the theta and tetrahedron graph. The \({\mathrm{SL}(2,\mathbb{C})}\) character variety of the fundamental group of the complement of a trivalent graph with E edges in S ³ is a Lagrangian subvariety of the Hitchin moduli space over the Riemann surface of genus g = E/3 + 1. For the theta and tetrahedron graph, we conjecture that the configuration of the character variety is locally determined by large color asymptotics of the quantum invariants of the trivalent graph in terms of complex Fenchel–Nielsen coordinates. Moreover, the q-holonomic difference equation of the quantum invariants provides the quantization of the character variety.     

Origin of regge symmetry for wigner coefficients ofLU(2)

Using general properties of the representations of unitary groups and their relations to representations of symmetric groups, the 3j symbol of the unitary unimodular group ?U(2) is written in terms of a 9j symbol of the unitary unimodular group ?U(J) withJ being the sum of the threej's. The result yields the Regge symmetry of the 3j symbol as a consequence of new relations between Wigner coefficients and special invariants of unitary groups on one hand and the association symmetry of the symmetric group on the other.     

Poisson Algebras of Block-Upper-Triangular Bilinear Forms and Braid Group Action

In this paper we study a quadratic Poisson algebra structure on the space of bilinear forms on ${\mathbb{C}^{N}}$ C N with the property that for any ${n, m \in \mathbb{N}}$ n , m ∈ N such that n m = N, the restriction of the Poisson algebra to the space of bilinear forms with a block-upper-triangular matrix composed from blocks of size ${m \times m}$ m × m is Poisson. We classify all central elements and characterise the Lie algebroid structure compatible with the Poisson algebra. We integrate this algebroid obtaining the corresponding groupoid of morphisms of block-upper-triangular bilinear forms. The groupoid elements automatically preserve the Poisson algebra. We then obtain the braid group action on the Poisson algebra as elementary generators within the groupoid. We discuss the affinisation and quantisation of this Poisson algebra, showing that in the case m = 1 the quantum affine algebra is the twisted q-Yangian for ${\mathfrak{o}_{n}}$ o n and for m = 2 is the twisted q-Yangian for ${(\mathfrak{sp}_{2n})}$ ( sp 2 n ) . We describe the quantum braid group action in these two examples and conjecture the form of this action for any m > 2. Finally, we give an R-matrix interpretation of our results and discuss the relation with Poisson–Lie groups.     

Decay of helicity in homogeneous turbulence

The self-similar relaxation of helicity in homogeneous turbulence has been considered taking into account integral invariants ∫ ₀ ^∞ r ^m 〈u(x)ω(x + r)〉 dr = I _m ^h (where ω = curlu and r = |r|). It has been shown that integral invariants with m = 3 for both helicity and energy are possible in addition to helical analogs of Loitsyanskii (m = 4) and Birkhoff-Saffman (m = 2) invariants associated with the conservation laws of momentum and angular momentum, respectively. Helicity always relaxes more rapidly than the energy. Its decay exponent is in the interval from ?3/2 to ?5/2 versus the interval from ?6/5 to ?10/7 for the energy.     

Local-Entire Cyclic Cocycles for Graded Quantum Field Nets

In a recent paper we studied general properties of super-KMS functionals on graded quantum dynamical systems coming from graded translation-covariant quantum field nets over ${\mathbb{R}}$ , and we carried out a detailed analysis of these objects on certain models of superconformal nets. In the present article, we show that these locally bounded functionals give rise to local-entire cyclic cocycles (generalized JLO cocycles) which are homotopy-invariant for a suitable class of perturbations of the dynamical system. Thus we can associate meaningful noncommutative geometric invariants to those graded quantum dynamical systems.     

Wavefunctions for topological quantum registers

We present explicit wavefunctions for quasi-hole excitations over a variety of non-abelian quantum Hall states: the Read-Rezayi states with k ? 3 clustering properties and a paired spin-singlet quantum Hall state. Quasi-holes over these states constitute a topological quantum register, which can be addressed by braiding quasi-holes. We obtain the braid properties by direct inspection of the quasi-hole wavefunctions. We establish that the braid properties for the paired spin-singlet state are those of ‘Fibonacci anyons’, and thus suitable for universal quantum computation. Our derivations in this paper rely on explicit computations in the parafermionic conformal field theories that underly these particular quantum Hall states.     

Non-traditional Approach to Quantum Computing

The N-qubit system characterized by an effective spin \(S = 2^{N - 1} - {1/2}\) is carried out in the representation of two coupled harmonic oscillators. It is shown that quantum computing results obtained with spinor algebra can be obtained also using the algebra of two coupled harmonic oscillators which is a convenient formalism, especially in the case of large number of qubits. In this formalism the non-abelian and abelian groups of the order of 16 related to one- and two-qubit systems were found. The structure of Cayley tables of those groups is different due to different commutation (anticommutation) relations for operators forming each group.     

Composite A,B,C,D-IRF-model invariants

In this work the introduction of generalized A,B,C,D interaction-round-a-face model invariants related to composite braid group representations will be proposed. The invariant polynomials are obtained in the framework of Witten's Chern-Simons theory summarizing recent works on link invariants. The primary intention is to present explicitly neglected results in the latter area and to outline in a pedagogical way the computation of a variety of known and new invariants. The close relationship of the topological interpretation of link invariants and the notion of generalized knot polynomials derived from integrable models in statistical mechanics is emphasized.     

Deformation Quantization on the Closure of Minimal Coadjoint Orbits

We consider a complex simple Lie algebra ${\mathfrak{g}}$ , with the action of its adjoint group. Among the three canonical nilpotent orbits under this action, the minimal orbit is the non zero orbit of smallest dimension. We are interested in equivariant deformation quantization: we construct ${\mathfrak{g}}$ -invariant star-products on the minimal orbit and on its closure, a singular algebraic variety. We shall make use of Hochschild homology and cohomology, of some results about the invariants of the classical groups, and of some interesting representations of simple Lie algebras. To the minimal orbit is associated a unique, completely prime two-sided ideal of the universal enveloping algebra ${{\rm U}(\mathfrak{g})}$ . This ideal is primitive and is called the Joseph ideal. We give explicit expressions for the generators of the Joseph ideal and compute the infinitesimal characters.     

Multiplicity-free Quantum 6j-Symbols for {U_q(\mathfrak{sl}_N)}

We conjecture a closed form expression for the simplest class of multiplicity-free quantum 6j-symbols for ${U_q(\mathfrak{sl}_N)}$ . The expression is a natural generalization of the quantum 6j-symbols for ${U_q(\mathfrak{sl}_2)}$ obtained by Kirillov and Reshetikhin. Our conjectured form enables computation of colored HOMFLY polynomials for various knots and links carrying arbitrary symmetric representations.     

Link Homologies and the Refined Topological Vertex

We establish a direct map between refined topological vertex and sl(N) homological invariants of the of Hopf link, which include Khovanov-Rozansky homology as a special case. This relation provides an exact answer for homological invariants of the Hopf link, whose components are colored by arbitrary representations of sl(N). At present, the mathematical formulation of such homological invariants is available only for the fundamental representation (the Khovanov-Rozansky theory) and the relation with the refined topological vertex should be useful for categorizing quantum group invariants associated with other representations (R ₁, R ₂). Our result is a first direct verification of a series of conjectures which identifies link homologies with the Hilbert space of BPS states in the presence of branes, where the physical interpretation of gradings is in terms of charges of the branes ending on Lagrangian branes.     

Quantum Mechanics on Manifolds and Topological Effects

A unique classification of the topological effects associated to quantum mechanics on manifolds is obtained on the basis of the invariance under diffeomorphisms and the realization of the Lie–Rinehart relations between the generators of the diffeomorphism group and the algebra of C ^∞ functions on the manifold. This leads to a unique (“Lie–Rinehart”) C ^*-algebra as observable algebra; its regular representations are shown to be locally Schroedinger and in one to one correspondence with the unitary representations of the fundamental group of the manifold. Therefore, in the absence of spin degrees of freedom and external fields, $ \pi_1{(\mathcal M)}$ appears as the only source of topological effects.     

Three-dimensional integrable models and associated tangle invariants

In this paper we show that the Boltzmann weights of the three-dimensional Baxter-Bazhanov model give representations of the braid group if some suitable spectral limits are taken. In the trigonometric case we classify all possible spectral limits which produce braid group representations. Furthermore, we prove that for some of them we get cyclotomic invariants of links and for others we obtain tangle invariants generalizing the cyclotomic ones.     

Localization of Unitary Braid Group Representations

Governed by locality, we explore a connection between unitary braid group representations associated to a unitary R-matrix and to a simple object in a unitary braided fusion category. Unitary R-matrices, namely unitary solutions to the Yang-Baxter equation, afford explicitly local unitary representations of braid groups. Inspired by topological quantum computation, we study whether or not it is possible to reassemble the irreducible summands appearing in the unitary braid group representations from a unitary braided fusion category with possibly different positive multiplicities to get representations that are uniformly equivalent to the ones from a unitary R-matrix. Such an equivalence will be called a localization of the unitary braid group representations. We show that the q = e ^πi/6 specialization of the unitary Jones representation of the braid groups can be localized by a unitary 9 × 9 R-matrix. Actually this Jones representation is the first one in a family of theories (SO(N), 2) for an odd prime N > 1, which are conjectured to be localizable. We formulate several general conjectures and discuss possible connections to physics and computer science.     

Spectral Measures for <Emphasis Type="Italic">G</Emphasis><Subscript>2</Subscript>

Spectral measures provide invariants for braided subfactors via fusion modules. In this paper we study joint spectral measures associated to the rank two Lie group G ₂, including the McKay graphs for the irreducible representations of G ₂ and its maximal torus, and fusion modules associated to all known G ₂ modular invariants.