A Kohno–Drinfeld Theorem for the Monodromy of Cyclotomic KZ Connections

We compute explicitly the monodromy representations of “cyclotomic” analogs of the Knizhnik–Zamolodchikov differential system. These are representations of the type B braid group B_n¹{B_n^1} . We show how the representations of the braid group B _n obtained using quantum groups and universal R-matrices may be enhanced to representations of B_n¹{B_n^1} using dynamical twists. Then, we show how these “algebraic” representations may be identified with the above “analytic” monodromy representations.     

Vertex operators in the conformal field theory on P 1 and monodromy representations of the braid group

Vertex operators (primary fields) are constructed for the conformal field theory on P ¹ by means of A ₁ ⁽¹⁾ modules. The commutation relations of vertex operators induce monodromy representations of the braid group on the spaces of vacuum expectations of compositions of vertex operators.     

On Metaplectic Modular Categories and Their Applications

For non-abelian simple objects in a unitary modular category, the density of their braid group representations, the #P-hard evaluation of their associated link invariants, and the BQP-completeness of their anyonic quantum computing models are closely related. We systematically study such properties of the non-abelian simple objects in the metaplectic modular categories SO(m)₂ for an odd integer m ≥ 3. The simple objects with quantum dimensions \({\sqrt{m}}\) have finite image braid group representations, and their link invariants are classically efficient to evaluate. We also provide classically efficient simulations of their braid group representations. These simulations of the braid group representations can be regarded as qudit generalizations of the Knill–Gottesmann theorem for the qubit case. The simple objects of dimension 2 give us a surprising result: while their braid group representations have finite images and are efficiently simulable classically after a generalized localization, their link invariants are #P-hard to evaluate exactly. We sharpen the #P-hardness by showing that any sufficiently accurate approximation of their associated link invariants is already #P-hard.     

Some monodromy representations of generalized braid groups

A flat connection on the trivial bundle over the complement inC ⁿ of the complexification of the system of the reflecting hyperplanes of theB _n,D _n Coxeter groups is built from a simple Lie algebra and its representation. The corresponding monodromy representations of the generalized braid groupsXB _n,XD _n are computed in the simplest case.     

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.     

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.     

Enhanced Gauge Symmetry and Braid Group Actions

 Enhanced gauge symmetry appears in Type II string theory (as well as F- and M-theory) compactified on Calabi–Yau manifolds containing exceptional divisors meeting in Dynkin configurations. It is shown that in many such cases, at enhanced symmetry points in moduli a braid group acts on the derived category of sheaves of the variety. This braid group covers the Weyl group of the enhanced symmetry algebra, which itself acts on the deformation space of the variety in a compatible way. Extensions of this result are given for nontrivial B-fields on K3 surfaces, explaining physical restrictions on the B-field, as well as for elliptic fibrations. The present point of view also gives new evidence for the enhanced gauge symmetry content in the case of a local A _2n -configuration in a threefold having global ℤ/2 monodromy. Received: 28 October 2002 / Accepted: 9 December 2002 Published online: 28 May 2003 Communicated by R.H. Dijkgraaf     

Random walks on the braid group B3 and magnetic translations in hyperbolic geometry

We study random walks on the three-strand braid group B₃, and in particular compute the drift, or average topological complexity of a random braid, as well as the probability of trivial entanglement. These results involve the study of magnetic random walks on hyperbolic graphs (hyperbolic Harper–Hofstadter problem), what enables to build a faithful representation of B₃ as generalized magnetic translation operators for the problem of a quantum particle on the hyperbolic plane.     

On geometric phases for soliton equations

This paper develops a new complex Hamiltonian structure forn-soliton solutions for a class of integrable equations such as the nonlinear Schrödinger, sine-Gordon and Korteweg-de Vries hierarchies of equations that yields, amongst other things, geometric phases in the sense of Hannay and Berry. For example, one of the possible soliton geometric phases is manifested by the well known phase shift that occurs for interacting solitons. The main new tools are complex angle representations that linearize the corresponding Hamiltonian flows on associated noncompact Jacobi varieties. This new structure is obtained by taking appropriate limits of the differential equations describing the class of quasi-periodic solutions. A method of asymptotic reduction of the angle representations is introduced for investigating soliton geometric phases that are related to the presence of monodromy at singularities in the space of parameters. In particular, the phase shift of interacting solitons can be expressed as an integral over a cycle on an associated Riemann surface. In this setting, soliton geometric asymptotics are constructed for studying geometric phases in the quantum case. The general approach is worked out in detail for the three specific hierarchies of equations mentioned. Some links with -functions, the braid group and geometric quantization are pointed out as well.Communicated by A. Jaffe     

Solvable RSOS models based on the dilute BWM algebra

In this paper we present representations of the recently introduced dilute Birman-Wenzl-Murakami algebra. These representations, labelled by the level-l B_n⁽¹⁾, C_n⁽¹⁾ and D_n⁽¹⁾ affine Lie algebras, are baxterized to yield solutions to the Yang-Baxter equation. The thus obtained critical solvable models are RSOS counterparts of the, respectively, D_n+1⁽²⁾, A_2n⁽²⁾ and B_n⁽¹⁾R-matrices of Bazhanov and Jimbo. For the D_n+1⁽²⁾ and B_n⁽¹⁾ algebras the RSOS models are new. An elliptic extension which solves the Yang-Baxter equation is given for all three series of dilute RSOS models.     

Uncovering Fractional Monodromy

The uncovering of the role of monodromy in integrable Hamiltonian fibrations has been one of the major advances in the study of integrable Hamiltonian systems in the past few decades: on one hand monodromy turned out to be the most fundamental obstruction to the existence of global action-angle coordinates while, on the other hand, it provided the correct classical analogue for the interpretation of the structure of quantum joint spectra. Fractional monodromy is a generalization of the concept of monodromy: instead of restricting our attention to the toric part of the fibration we extend our scope to also consider singular fibres. In this paper we analyze fractional monodromy for n ₁:(?n ₂) resonant Hamiltonian systems with n ₁, n ₂ coprime natural numbers. We consider, in particular, systems that for n ₁, n ₂ > 1 contain one-parameter families of singular fibres which are ‘curled tori’. We simplify the geometry of the fibration by passing to an appropriate branched covering. In the branched covering the curled tori and their neighborhood become untwisted thus simplifying the geometry of the fibration: we essentially obtain the same type of generalized monodromy independently of n ₁, n ₂. Fractional monodromy is then recovered by pushing the results obtained in the branched covering back to the original system.     

From chiral to two-dimensional Wess-Zumino-Novikov-Witten model via quantum gauge group

We study the canonical quantization of the SU(n) WZNW model. Decoupling the chiral dynamics requires an extended state space including left and right monodromies as independent variables. In the simplest (n = 2) case we explicitly show that the zero modes of the monodromy extended SU(2) WZNW model give rise to a quantum group gauge theory in a finite-dimensional Fock space. We define the subspace of U_q(sl(2)) ⊗ U_q(sl(2))-invariant vectors on which the monodromy invariance is also restored and construct the physical space applying a generalized cohomology condition.     

Scaling in the quantum Hall effect regime in n-InGaAs/GaAs nanostructures

The longitudinal ρ _xx (B) and Hall ρ _xy (B) magnetoresistances are investigated experimentally in the integer quantum Hall effect (QHE) regime in n-InGaAs/GaAs double quantum well nanostructures in the range of magnetic fields B = (0–16) T and temperatures T = (0.05–70) K before and after IR illumination. The results are evaluated within the scaling hypothesis with regard to electron-electron interaction.     

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.