Knot invariants associated with a particularN→∞ continuous limit of the Baxter-Bazhanov model

First we briefly recall the definition of the three-dimensional Baxter-Bazhanov lattice model. The spins of this model are elements ofZ _N and theR-matrix is associated to the algebraU _q sl(n) ifq is a primitiveNth root of unity. Then we construct a particularN limit of the model, in which it is meaningful to interpret the spins as elements ofR and which gives the free Gaussian boson model. Finally, we study special limits of the rapidity variables in which we obtain braid group representations and we show that forn odd the associated knot invariants are given by the inverse of products of Alexander polynomials, evaluated at certain roots of unity.     

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.     

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.     

On the geometry of some unitary Riemann surface braid group representations and Laughlin-type wave functions

In this note we construct the simplest unitary Riemann surface braid group representations geometrically by means of stable holomorphic vector bundles over complex tori and the prime form on Riemann surfaces. Generalised Laughlin wave functions are then introduced. The genus one case is discussed in some detail also with the help of noncommutative geometric tools, and an application of Fourier–Mukai–Nahm techniques is also given, explaining the emergence of an intriguing Riemann surface braid group duality.     

On the structure of unitary conformal field theory II: Representation theoretic approach

Two-dimensional, unitary rational conformal field theory is studied from the point of view of the representation theory of chiral algebras. Chiral algebras are equipped with a family of co-multiplications which serve to define tensor product representations. Chiral vertices arise as Clebsch-Gordan operators from tensor product representations to irreducible subrepresentations of a chiral algebra. The algebra of chiral vertices is studied and shown to give rise to representations of the braid groups determined by Yang-Baxter (braid) matrices. Chiral fusion is analyzed. It is shown that the braid- and fusion matrices determine invariants of knots and links. Connections between the representation theories of chiral algebras and of quantum groups are sketched. Finally, it is shown how the local fields of a conformal field theory can be reconstructed from the chiral vertices of two chiral algebras.     

Topological Invariants in Braid Theory

Many invariants of knots and links have their counterparts in braid theory. Often, these invariants are most easily calculated using braids. A braid is a set of n strings stretching between two parallel planes. This review demonstrates how integrals over the braid path can yield topological invariants. The simplest such invariant is the winding number – the net number of times two strings in a braid wrap about each other. But other, higher-order invariants exist. The mathematical literature on these invariants usually employs techniques from algebraic topology that may be unfamiliar to physicists and mathematicians in other disciplines. The primary goal of this paper is to introduce higher-order invariants using only elementary differential geometry.Some of the higher-order quantities can be found directly by searching for closed one-forms. However, the Kontsevich integral provides a more general route. This integral gives a formal sum of all finite order topological invariants. We describe the Kontsevich integral, and prove that it is invariant to deformations of the braid.Some of the higher-order invariants can be used to generate Hamiltonian dynamics of n particles in the plane. The invariants are expressed as complex numbers; but only the real part gives interesting topological information. Rather than ignoring the imaginary part, we can use it as a Hamiltonian. For n = 2, this will be the Hamiltonian for point vortex motion in the plane. The Hamiltonian for n = 3 generates more complicated motions.     

Knitting Ansatz and Solutions to Yang-Baxter Equation

We suggest a new method,named knitting ansatz,to generate solutions to Yang-Baxter equation with lower dimensional representations of braid group.To support our ansatz,we work out an example of a new 16×16 R-matrix constructed along this idea,with two 4×4 braid group representations of familiar 6-vertex type with different q-parameters.     

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.     

Fusion and braiding in W-algebra extended conformal theories

We define the chiral conformal blocks of integer-spin extended (W-algebra) conformal theories by the fusion of elementary ones. The braid group representation matrices which realize the exchange algebra are computed. They are shown to coincide with the Boltzmann weights — in a certain limit of the spectral parameter — of the critical face models of Jimbo et al. In the unitary cases, where the extended conformal theories can be realized as cosets , we relate the braiding matrices of the former to those of the WZW models. In this article we restrict ourselves to the case corresponding to symmetric tensor representations of A_n.     

Knots,links, braids and exactly solvable models in statistical mechanics

We present a general method to construct the sequence of new link polynomials and its two variable extension from exactly solvable models in statistical mechanics. First, we find representations of the braid group from the Boltzmann weights of the exactly solvable models. Second, we give the Markov traces associated with new braid group representations and using them construct new link polynomials. Third, we extend the theory into a two-variable version of the new link polynomials. Throughout the paper, we emphasize the essential roles played by the exactly solvable models and the underlying Yang-Baxter relation.     

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.     

On the two-vector invariants of point groups

The paper is devoted to group-theoretical analysis of a function of two vectors, invariant under some point group (special attention is paid to the O_h group). In particular, the invariants in the direct product of spaces transforming according to the lth and l'th irreducible representations of the rotation group are studied. A compact formula determining the number of such invariants for the group O_h is found. It is shown that all possible invariants in the considered product space can be constructed from all possible scalar products of vector functions of both vectors transforming according to complex conjugate irreducible representations. The addition theorem for these functions is proved.     

New implementation of composite fermions in terms of subgroups of a braid group

The new implementation of composite fermions and more generally — of composite anyons is formulated, exploiting one-dimensional unitary representations of appropriately constructed subgroups of the full braid group. The nature of hypothetical fluxes attached to the Jain's composite fermions is explained via additional cyclotron trajectory loops consistently with the braid subgroup structure. It is demonstrated that composite fermions are proper 2D particles (not an auxiliary construction), but associated with braid subgroups instead of the full braid group.     

Construction and classification of indecomposable finite-dimensional representations of the homogeneous Galilei group

We discuss finite-dimensional representations of the homogeneous Galilei group which, when restricted to its subgroup SO(3), are decomposed to spin 0, 1/2 and 1 representations. In particular we explain how these representations were obtained in our paper (M. de Montigny et al.: J. Phys. A39 (2006) 9365) via reduction of the classification problem to a matrix one admitting exact solutions, and via contraction of the corresponding representations of the Lorentz group. Finally, for discussed representations we derive all functional invariants.     

The physical meaning of the de Sitter invariants

We study the Lie algebras of the covariant representations transforming the matter fields under the de Sitter isometries. We point out that the Casimir operators of these representations can be written in closed forms and we deduce how their eigenvalues depend on the field’s rest energy and spin. For the scalar, vector and Dirac fields, which have well-defined field equations, we express these eigenvalues in terms of mass and spin obtaining thus the principal invariants of the theory of free fields on the de Sitter spacetime. We show that in the flat limit we recover the corresponding invariants of the Wigner irreducible representations of the Poincaré group.     

Link Polynomials and Hamiltonian Corresponding to Braid Group Representations of ZN Model

The more general braid group representations of ZZ_N model are given. Topologically invari-ant link polynomials related to the model are constructed. Here we present the simplest Hamiltonian with arbitrary coefficients of 1D quantum chain.     

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.