Quantized affine Lie algebras and diagonalization of braid generators

Let U _q be a quantized affine Lie algebra. It is proven that the universal R-matrix R of U _q satisfies the celebrated conjugation relationR ⁺ =TR withT the usual twist map. As applications, the braid generator is shown to be diagonalizable on arbitrary tensor product modules of integrable irreducible highest weight U _q -module and a spectral decomposition formula for the braid generator is obtained which is the generalization of Reshetikhin and Gould forms to the present affine case. Casimir invariants are constructed and their eigenvalues computed by means of the spectral decomposition formula. As a by-product, an interesting identity is found.     

Diagonalization of the braid generator on unitary irreps of quantum supergroups

It is shown that the braid generator associated with the universalR-matrix is diagonalizable on all unitary representations of quantum supergroups. An example is considered using U _q (gl(2|1)) and a family of eight-dimensional typical representations.     

Link invariants of finite type and perturbation theory

The Vassiliev-Gusarov link invariants of finite type are known to be closely related to perturbation theory for Chern-Simons theory. In order to clarify the perturbative nature of such link invariants, we introduce an algebra V _x containing elements g _i satisfying the usual braid group relations and elements a _i satisfying g _i–g _infi ^sup-1 =a _i, where is a formal variable that may be regarded as measuring the failure of g _infi ^sup2 to equal 1. Topologically, the elements a _i signify intersections. We show that a large class of link invariants of finite type are in one-to-one correspondence with homogeneous Markov traces on V _x. We sketch a possible application of link invariants of finite type to a manifestly diffeomorphisminvariant perturbation theory for quantum gravity in the loop representation.     

Lifschitz singularities for periodic operators plus random potentials

UsingX-bounding (lower bounds by Laplacians with mixed boundary conditions and discrete analogs), we obtain the Lifschitz exponent at the bottom of the spectrum for random operators of typeH =T+V , withT a (periodic) generator of a positivity-preserving semigroup, extending results by Kirsch and Simon.     

Braided matrix structure of the Sklyanin algebra and of the quantum Lorentz group

Braided groups and braided matrices are novel algebraic structures living in braided or quasitensor categories. As such they are a generalization of super-groups and super-matrices to the case of braid statistics. Here we construct braided group versions of the standard quantum groupsU _q (g). They have the same FRT generatorsl ^± but a matrix braided-coproductL=LL, whereL=l ⁺ Sl ^–, and are self-dual. As an application, the degenerate Sklyanin algebra is shown to be isomorphic to the braided matricesBM _q(2); it is a braided-commutative bialgebra in a braided category. As a second application, we show that the quantum doubleD(U _q (sl ₂)) (also known as the quantum Lorentz group) is the semidirect product as an algebra of two copies ofU _q (sl ₂), and also a semidirect product as a coalgebra if we use braid statistics. We find various results of this type for the doubles of general quantum groups and their semi-classical limits as doubles of the Lie algebras of Poisson Lie groups.     

Partial waves of a general off-shell two-body Coulomb-like T-matrix

On the basis of the Gell-Mann — Goldberger two-potential formalism we investigate the partial waves of an off-shell two-body T-matrix in the case of a general Coulomb-like potentialV=V _C +V _S . The regular kernelt _SC,l determining thel-th partial wave of the short-range partT _SC,l of the T-matrix is the solution of the equationt _SC,l =V _S,l +V _S,l G _C,l t _SC,l . The Lippmann-Schwinger operator of this equation formed by the short-range part of the potential and the pure Coulomb Green's operator is shown to be compact under very general assumptions on the potentialV _S admitting potentials vanishing in the coordinate representation liker ^–1– (>0) in the infinity. The special case of differentiable and analytic potentialsV _S,l (p,p) is considered in particular. The results are used to discuss in full generality the on-shell singularities of Coulomb-like T-matrices and wave functions and to investigate the singular integrals that occur in the Faddeev equations for Coulomb-like interactions.     

Monopoles,braid groups,and the dirac operator

Using the relation between the space of rational functions on , the space ofSU(2)-monopoles on ³, and the classifying space of the braid group, see [10], we show how the index bundle of the family of real Dirac operators coupled toSU(2)-monopoles can be described using permutation representations of Artin's braid groups. We also show how this implies the existence of a pair consisting of a gauge fieldA and a Higgs field on ³ whose corresponding Dirac equation has an arbitrarily large dimensional space of solutions.The first author was supported by a grant from the NSF     

The equilibrium thermodynamics of a spin-boson model

We consider the equilibrium thermodynamics of a Dicke-type model forN identical spins of arbitrary magnitude interacting linearly and homogeneously with a boson field in a volumeV _N, in the limitN,V _N, withN/V _N=const. The system exhibits a second-order phase transition; complete information on the spin polarizations and their correlations is obtained. The proofs use a general result on the free energy of quantum spin systems based on the large deviation principle and the Berezin-Lieb inequalities.     

Microscopic Calculation of the Dielectric Susceptibility Tensor for Coulomb Fluids

In a Coulomb fluid confined to a domain V, the dielectric susceptibility tensor _V depends on the shape of V, even in the thermodynamic V limit. This paper deals with the classical two-dimensional one-component plasma formulated in an elliptic V-domain, equilibrium statistical mechanics is used. For the dimensionless coupling constant =even positive integer, the mapping of the plasma onto a discrete one-dimensional anticommuting-field theory provides a new sum rule. This sum rule confirms the limiting value of _V predicted by macroscopic electrostatics and gives a finite-size correction term to _V.     

Induced modules for vertex operator algebras

For a vertex operator algebraV and a vertex operator subalgebraV which is invariant under an automorphismg ofV of finite order, we introduce ag-twisted induction functor from the category ofg-twistedV-modules to the category ofg-twistedV-modules. This functor satisfies the Frobenius reciprocity and transitivity. The results are illustrated withV being theg-invariants in simpleV orV beingg-rational.The first author was supported by NSF grant DMS-9303374 and a research grant from the Committee on Research, UC Santa Cruz.The second author was supported by NSF grant DMS-9401389.     

Integrable graded manifolds and nonlinear equations

A method is proposed for the classification of integrable embeddings of (2+2)-dimensional supermanifoldsV _2|2 into an enveloping superspace supplied with the structure of a Lie superalgebra. The approach is first applied to the even part of the scheme, i.e. for the embeddings of 2-dimensional manifoldsV ₂ into Riemannian or non-Riemannian enveloping space. The general consideration is also illustrated by the example of superspaces supplied with the structure of the series sl(n, n+1), whose integrable supermanifolds are described by supersymmetrical 2-dimensional Toda lattice type equations. In particular, forn=1 they are described by the supersymmetrical Liouville and Sine-Gordon equations.     

Analyticity in Hubbard Models

The Hubbard model describes a lattice system of quantum particles with local (on-site) interactions. Its free energy is analytic when t is small, or t ²/U is small; here, is the inverse temperature, U the on-site repulsion, and t the hopping coefficient. For more general models with Hamiltonian H=V+T where V involves local terms only, the free energy is analytic when T is small, irrespective of V. There exists a unique Gibbs state showing exponential decay of spatial correlations. These properties are rigorously established in this paper.     

Field operators for anyons and plektons

Given its superselection sectors with non-abelian braid group statistics, we extend the algebraA of local observables into an algebra containing localized intertwiner fields which carry the superselection charges. The construction of the inner degrees of freedom, as well as the study of their transformation properties (quantum symmetry), are entirely in terms of the superselection structure of the observables. As a novel and characteristic feature for braid group statistics, Clebsch-Gordan and commutation coefficients generically take values in the algebra of symmetry operators, much as it is the case with quasi-Hopf symmetry.A, , and are allC ^* algebras, i.e. represented by bounded operators on a Hilbert space with positive metric.     

Characterizing invariants for local extensions of current algebras

ParisA of local quantum field theories are studied, whereA is a chiral conformal quantum field theory and is a local extension, either chiral or two-dimensional. The local correlation functions of fields from have an expansion with respect toA into conformal blocks, which are non-local in general. Two methods of computing characteristic invariant ratios of structure constants in these expansions are compared: (a) by constructing the monodromy representation of the braid group in the space of solutions of the Knizhnik-Zamolodchikov differential equation, and (b) by an analysis of the local subfactors associated with the extension with methods from operator algebra (Jones theory) and algebraic quantum field theory. Both approaches apply also to the reverse problem: the characterization and (in principle) classification of local extensions of a given theory.     

Production of a decoupled neutral gauge boson at the SSC

We consider an extension of the standard model with an extra neutral gauge bosonV which does not couple to the usual fermions or mix with the standardZ boson. A new fermion, which is either a color-triplet (Q) or a color-singlet (L), and a singlet Higgs scalar (X) are generic particles required for the extension group.V can be produced copiously by the process . We find 0760 0712 V 3 that there will be at least a few hundredV's produced at the SSC per year if the masses ofV andQ are less than 1 and 0.3 TeV respectively. Although the mixing of the standard and new Higgs scalars is small, the decay of the standard model Higgs scalar,H ₁2V, is comparable to the corresponding standard processes,H ₁2Z. A heavy standard model Higgs may open an important channel for the study of a new neutral gauge boson.     

Conformal field algebras with quantum symmetry from the theory of superselection sectors

According to the theory of superselection sectors of Doplicher, Haag, and Roberts, field operators which make transitions between different superselection sectors—i.e. different irreducible representations of the observable algebra—are to be constructed by adjoining localized endomorphisms to the algebra of local observables. We find the relevant endomorphisms of the chiral algebra of observables in the minimal conformal model with central chargec=1/2 (Ising model). We show by explicit and elementary construction how they determine a representation of the braid groupB which is associated with a Temperley-Lieb-Jones algebra. We recover fusion rules, and compute the quantum dimensions of the superselection sectors. We exhibit a field algebra which is quantum group covariant and acts in the Hilbert space of physical states. It obeys local braid relations in an appropriate weak sense.     

Reweighting of the form factors in exclusive $B \rightarrow X \ell \nu_{\ell}$ decays

A form factor reweighting technique has been elaborated to permit relatively easy comparisons between different form factor models applied to exclusive decays. The software tool developed for this purpose is described. It can be used with any event generator, three of which were used in this work: ISGW2, PHSP and FLATQ2, a new powerful generator. The software tool allows for an easy and reliable implementation of any form factor model. The tool has been fully validated with the ISGW2 form factor hypothesis. The results of our present studies indicate that the combined use of the FLATQ2 generator and the form factor reweighting tool should play a very important role in future exclusive |V _ub | measurements, with largely reduced errors.Received: 28 May 2004, Revised: 23 July 2004, Published online: 3 November 2004     

Scalar Fields in Multidimensional Gravity. No-Hair and Other No-Go Theorems

Global properties of static, spherically symmetric configurations with scalar fields of sigma-model type with arbitrary potentials are studied in D dimensions, including models where the space-time contains multiple internal factor spaces. The latter are assumed to be Einstein spaces, not necessarily Ricci-flat, and the potential V includes a contribution from their curvatures. The following results generalize those known in four dimensions: (A) a no-hair theorem on the nonexistence, in case V 0, of asymptotically flat black holes with varying scalar fields or moduli fields outside the event horizon; (B) nonexistence of particlelike solutions in field models with V 0; (C) nonexistence of wormhole solutions under very general conditions; (D) a restriction on possible global causal structures (represented by Carter-Penrose diagrams). The list of structures in all models under consideration is the same as is known for vacuum with a cosmological constant in general relativity: Minkowski (or AdS), Schwarzschild, de Sitter and Schwarzschild – de Sitter, and horizons which bound a static region are always simple. The results are applicable to various Kaluza-Klein, supergravity and stringy models with multiple dilaton and moduli fields.     

Mass-Proper Time Uncertainty Relation in a Manifestly Covariant Relativistic Statistical Mechanics

We prove the uncertainty relation T _V m2/c ², which is realized on a statistical mechanical level for an ensemble of events in (1+D)-dimensional spacetime with motion parameterized by an invariant proper time , where T _V is the average passage interval in for the events which pass through a small (typical) (1+D)-volume V, and m is the dispersion of mass around its on-shell value in such an ensemble. We show that a linear mass spectrum is a completely general property of a (1+D)-dimensional off-shell theory.On sabbatical leave from School of Physics and Astronomy, Tel Aviv Uniersity, Ramat Aviv, Israel. Also at Department of Physics, Bar-Ilan University, Ramat-Gan, Israel     

Noncommutative Independence from the Braid Group $${\mathbb{B}_{\infty}}$$

We introduce ‘braidability’ as a new symmetry for infinite sequences of noncommutative random variables related to representations of the braid group \({\mathbb{B}_{\infty}}\) . It provides an extension of exchangeability which is tied to the symmetric group \({\mathbb{S}_{\infty}}\) . Our key result is that braidability implies spreadability and thus conditional independence, according to the noncommutative extended de Finetti theorem [Kös08]. This endows the braid groups \({\mathbb{B}_{n}}\) with a new intrinsic (quantum) probabilistic interpretation. We underline this interpretation by a braided extension of the Hewitt-Savage Zero-One Law. Furthermore we use the concept of product representations of endomorphisms [Goh04] with respect to certain Galois type towers of fixed point algebras to show that braidability produces triangular towers of commuting squares and noncommutative Bernoulli shifts. As a specific case we study the left regular representation of \({\mathbb{B}_{\infty}}\) and the irreducible subfactor with infinite Jones index in the non-hyperfinite I I ₁-factor L \({(\mathbb{B}_{\infty})}\) related to it. Our investigations reveal a new presentation of the braid group \({\mathbb{B}_{\infty}}\) , the ‘square root of free generator presentation’ \({\mathbb{F}^{1/2}_{\infty}}\) . These new generators give rise to braidability while the squares of them yield a free family. Hence our results provide another facet of the strong connection between subfactors and free probability theory [GJS07]; and we speculate about braidability as an extension of (amalgamated) freeness on the combinatorial level.