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arXiv1997

Bargmann representations for deformed harmonic oscillators

Generalizing the case of the usual harmonic oscillator, we look for Bargmann representations corresponding to deformed harmonic oscillators. Deformed harmonic oscillator algebras are generated by four operators $a, a^\dagger, N$ and the unity 1 such as $[a,N] = a, [a^\dagger,N] = -a^\dagger$, $a^\dagger a = ψ(N)$ and $aa^\dagger =ψ(N+1)$. We discuss the conditions of existence of a scalar product expressed with a true integral on the space spanned by the eigenstates of $a$ (or $a^\dagger$). We give various examples, in particular we consider functions $ψ$ that are linear combinations of $q^N$, $q^{-N}$ and unity and that correspond to q-oscillators with Fock-representations or with non-Fock-representations.

q-alg↗
arXiv1995

Knots and Numbers in $ϕ^4$ Theory to 7 Loops and Beyond

We evaluate all the primitive divergences contributing to the 7--loop $β$\/--function of $ϕ^4$ theory, i.e.\ all 59 diagrams that are free of subdivergences and hence give scheme--independent contributions. Guided by the association of diagrams with knots, we obtain analytical results for 56 diagrams. The remaining three diagrams, associated with the knots $10_{124}$, $10_{139}$, and $10_{152}$, are evaluated numerically, to 10 sf. Only one satellite knot with 11 crossings is encountered and the transcendental number associated with it is found. Thus we achieve an analytical result for the 6--loop contributions, and a numerical result at 7 loops that is accurate to one part in $10^{11}$. The series of `zig--zag' counterterms, $\{6ζ_3,\,20ζ_5,\, \frac{441}{8}ζ_7,\,168ζ_9,\,\ldots\}$, previously known for $n=3,4,5,6$ loops, is evaluated to 10 loops, corresponding to 17 crossings, revealing that the $n$\/--loop zig--zag term is $4C_{n-1} \sum_{p>0}\frac{(-1)^{p n - n}}{p^{2n-3}}$, where $C_n=\frac{1}{n+1}{2n \choose n}$ are the Catalan numbers, familiar in knot theory. The investigations reported here entailed intensive use of REDUCE, to generate ${\rm O}(10^4)$ lines of code for multiple precision FORTRAN computations, enabled by Bailey's MPFUN routines, running for ${\rm O}(10^3)$ CPUhours on DecAlpha machines.

hep-ph↗
arXiv1995

A Link Invariant from Quantum Dilogarithm

The link invariant, arising from the cyclic quantum dilogarithm via the particular $R$-matrix construction, is proved to coincide with the invariant of triangulated links in $S^3$ introduced in R.M. Kashaev, Mod. Phys. Lett. A, Vol.9 No.40 (1994) 3757. The obtained invariant, like Alexander-Conway polynomial, vanishes on disjoint union of links. The $R$-matrix can be considered as the cyclic analog of the universal $R$-matrix associated with $U_q(sl(2))$ algebra.

q-alg↗
arXiv1994

Representations of Composite Braids and Invariants for Mutant Knots and Links in Chern-Simons Field Theories

We show that any of the new knot invariants obtained from Chern-Simons theory based on an arbitrary non-abelian gauge group do not distinguish isotopically inequivalent mutant knots and links. In an attempt to distinguish these knots and links, we study Murakami (symmetrized version) $r$-strand composite braids. Salient features of the theory of such composite braids are presented. Representations of generators for these braids are obtained by exploiting properties of Hilbert spaces associated with the correlators of Wess-Zumino conformal field theories. The $r$-composite invariants for the knots are given by the sum of elementary Chern-Simons invariants associated with the irreducible representations in the product of $r$ representations (allowed by the fusion rules of the corresponding Wess-Zumino conformal field theory) placed on the $r$ individual strands of the composite braid. On the other hand, composite invariants for links are given by a weighted sum of elementary multicoloured Chern-Simons invariants. Some mutant links can be distinguished through the composite invariants, but mutant knots do not share this property. The results, though developed in detail within the framework of $SU(2)$ Chern-Simons theory are valid for any other non-abelian gauge group.

hep-th↗
arXiv1995

The Master Field for Large $N$ Matrix Models and Quantum Groups

In recent works by Singer, Douglas and Gopakumar and Gross an application of results of Voiculescu from non-commutative probability theory to constructions of the master field for large $N$ matrix field theories have been suggested. In this note we consider interrelations between the master field and quantum groups. We define the master field algebra and observe that it is isomorphic to the algebra of functions on the quantum group $SU_q(2)$ for $q=0$. The master field becomes a central element of the quantum group Hopf algebra. The quantum Haar measure on the $SU_q(2)$ for any $q$ gives the Wigner semicircle distribution for the master field. Coherent states on $SU_q(2)$ become coherent states in the master field theory.

hep-th↗
arXiv1996

w_{\infty} and sl_q(2) Algebras in the Landau Problem and Chern-Simons Theory on a Torus

We discuss $w_\infty$ and $sl_q(2)$ symmetries in Chern-Simons theory and Landau problem on a torus. It is shown that when the coefficient of the Chern-Simons term, or when the total flux passing through the torus is a rational number, there exist in general two $w_\infty$ and two $sl_q(2)$ algebras, instead of one set each discussed in the literature. The general wavefunctions for the Landau problem with rational total flux is also presented.

hep-th↗
arXiv1995

Lie-Poisson groups and the Miura transformation

We point out that the recent proof of the Kupershmidt-Wilson theorem by Cheng and Mas-Ramos is underpinned by the Lie-Poisson property of the second Gel'fand-Dickey bracket. The supersymmetric Kupershmidt-Wilson theorem is also proved along these same lines. Finally we comment on the possible repercussions in the problem of the coproduct for W-algebras.

q-alg↗
arXiv1995

Exact Bethe ansatz solution of nonultralocal quantum mKdV model

A lattice regularized Lax operator for the nonultralocal modified Korteweg de Vries (mKdV) equation is proposed at the quantum level with the basic operators satisfying a $q$-deformed braided algebra. Finding further the associated quantum $R$ and $Z$-matrices the exact integrability of the model is proved through the braided quantum Yang--Baxter equation, a suitably generalized equation for the nonultralocal models. Using the algebraic Bethe ansatz the eigenvalue problem of the quantum mKdV model is exactly solved and its connection with the spin-$\ha$ XXZ chain is established, facilitating the investigation of the corresponding conformal properties.

hep-th↗
arXiv1995

The universal R-matrix for the Jordanian deformation of sl(2), and the contracted forms of so(4)

We introduce a universal R matrix for the Jordanian deformation of $\U{ \sl(2)}$. Using $\Uh{\so(4)}=\Uh{\sl(2)} \oplus {\rm U}_{-h}(\sl(2))$, we obtain the universal R matrix for $\Uh{\so(4)}$. Applying the graded contractions on the universal R matrix of $\Uh{\so(4)}$, we show that there exist three distinct R matrices for all of the contracted algebras. It is shown that $\Uh{\sl(2)}$, $\Uh{\so(4)}$, and all of these contracted algebras are triangular.

q-alg↗
arXiv1996

Quantum Deformation of igl(n) Algebra on Quantum Space

We study quantum deformed $gl(n)$ and $igl(n)$ algebras on a quantum space discussing multi-parametric extension. We realize elements of deformed $gl(n)$ and $igl(n)$ algebras by a quantum fermionic space. We investigate a map between deformed $igl(2)$ algebras of our basis and other basis.

q-alg↗
arXiv1996

(2+1) null-plane quantum Poincaré group from a factorized universal $R$-matrix

The non-standard (Jordanian) quantum deformations of $so(2,2)$ and (2+1) Poincaré algebras are constructed by starting from a quantum $sl(2,\R)$ basis such that simple factorized expressions for their corresponding universal $R$-matrices are obtained. As an application, the null-plane quantum (2+1) Poincaré Poisson-Lie group is quantized by following the FRT prescription. Matrix and differential representations of this null-plane deformation are presented, and the influence of the choice of the basis in the resultant $q$-Schrödinger equation governing the deformed null plane evolution is commented.

q-alg↗
arXiv1997

R-deformed Heisenberg algebra

It is shown that the deformed Heisenberg algebra involving the reflection operator R (R-deformed Heisenberg algebra) has finite-dimensional representations which are equivalent to representations of paragrassmann algebra with a special differentiation operator. Guon-like form of the algebra, related to the generalized statistics, is found. Some applications of revealed representations of the R-deformed Heisenberg algebra are discussed in the context of OSp(2|2) supersymmetry. It is shown that these representations can be employed for realizing (2+1)-dimensional supersymmetry. They give also a possibility to construct a universal spinor set of linear differential equations describing either fractional spin fields (anyons) or ordinary integer and half-integer spin fields in 2+1 dimensions.

hep-th↗
arXiv1997

Tetrahedron Reflection Equation

Reflection equation for the scattering of lines moving in half-plane is obtained. The corresponding geometric picture is related with configurations of half-planes touching the boundary plane in 2+1 dimensions. This equation can be obtained as an additional to the tetrahedron equation consistency condition for a modified Zamolodchikov algebra.

hep-th↗
arXiv1997

Fusion rules for Quantum Transfer Matrices as a Dynamical System on Grassmann Manifolds

We show that the set of transfer matrices of an arbitrary fusion type for an integrable quantum model obey these bilinear functional relations, which are identified with an integrable dynamical system on a Grassmann manifold (higher Hirota equation). The bilinear relations were previously known for a particular class of transfer matrices corresponding to rectangular Young diagrams. We extend this result for general Young diagrams. A general solution of the bilinear equations is presented.

solv-int↗
arXiv1997

Drinfeld twist for quantum $su(2)$ in the adjoint representation

We give a detailed description of the adjoint representation of Drinfeld's twist element, as well as of its coproduct, for $su_{q}(2)$. We also discuss, as applications, the computation of the universal R-matrix in this representation and the problem of symmetrization of identical-particle states with quantum $su(2)$ symmetry.

q-alg↗