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arXiv1994

On the operator content of nilpotent orbifold models

Let $V$ be a simple vertex operator algebra and $G$ be a finite nilpotent group of automorphisms of $V.$ We prove the following in this paper: (1) There is a Galois correspondence between subgroups of $G$ and the vertex operator subalgebras of $V$ which contain $V^G$ given by the map $H\mapsto V^H.$ (2) Assume that for every G\in G$ there is unique simple $g$-twisted $V$-module $M(g).$ Then there exists a Hochschild 3-cocycle $α$ on the integral group $Z[G]$ such that there is an equivalence of categories between $V^G$-module category (whose objects are $V^G$-submodules of direct sums of copies of $\oplus_{g\in G}M(g),$ and whose morphisms are $V^G$-module homomorphisms) and the module category for the twisted quantum double $D_α(G)$ associated to $α.$

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arXiv1994

Differential Geometry on Hopf Algebras and Quantum Groups (Ph.D. Thesis)

The differential geometry on a Hopf algebra is constructed, by using the basic axioms of Hopf algebras and noncommutative differential geometry. The space of generalized derivations on a Hopf algebra of functions is presented via the smash product, and used to define and discuss quantum Lie algebras and their properties. The Cartan calculus of the exterior derivative, Lie derivative, and inner derivation is found for both the universal and general differential calculi of an arbitrary Hopf algebra, and, by restricting to the quasitriangular case and using the numerical R-matrix formalism, the aforementioned structures for quantum groups are determined.

hep-th↗
arXiv1994

Non--Commutative (Quantum) Probability, Master Fields and Stochastic Bosonization

In this report we discuss some results of non--commutative (quantum) probability theory relating the various notions of statistical independence and the associated quantum central limit theorems to different aspects of mathematics and physics including: $q$--deformed and free central limit theorems; the description of the master (i.e. central limit) field in matrix models along the recent Singer suggestion to relate it to Voiculescu's results on the freeness of the large $N$ limit of random matrices; quantum stochastic differential equations for the gauge master field in QCD; the theory of stochastic limits of quantum fields and its applications to stochastic bosonization of Fermi fields in any dimensions; new structures in QED such as a nonlinear modification of the Wigner semicircle law and the interacting Fock space: a natural explicit example of a self--interacting quantum field which exhibits the non crossing diagrams of the Wigner semicircle law.

hep-th↗
arXiv1995

Spinor construction of the c = 1/2 minimal model

The usual spinor construction from one fermion yields four irreducible representations of the Virasoro algebra with central charge $c = 1/2$. The Neveu-Schwarz (NS) sector is the direct sum of an $h = 0$ and an $h = 1/2$ module, and the Ramond (R) sector is the direct sum of two copies of an $h = 1/16$ module. In addition to the fundamental fermions, which represent a Clifford algebra, and the Virasoro operators, there are infinitely many other vertex operators, in one-to-one correspondence with the vectors (states) in the NS sector. These give the NS sector the structure of a Vertex Operator SuperAlgebra, and the R sector the structure of a ${\bold Z}_2$-twisted module for that VOSA. Keeping both copies of the $h = 1/16$ modules in the R sector, we can define intertwining operators in one-to-one correspondence with the states in the R sector such that the usual Ising fusion rules for just three modules are replaced by a rule given by the group ${\bold Z}_4$. The main objective is to find a generalization of the VOSA Jacobi-Cauchy identity which is satisfied by these intertwining operators. There are several novel features of this new ``Matrix'' Jacobi-Cauchy Identity (MJCI), most of which come from the fact that correlation functions made from two intertwiners are hypergeometric functions. In order to relate and rationalize the correlation functions we use the Kummer quadratic transformation formulas, lifting the functions to a four-sheeted covering, branched over the usual three poles, where the Cauchy residue theorem can be applied. The six possible poles on the cover give six terms in the MJCI. Furthermore, we organize those functions into $2\times 4$ matrices and find the $2\times 2$ (fusion and braiding) matrices which relate them at the six poles. These results for intertwiners

hep-th↗
arXiv1995

Théorie de Liouville et gravitation quantique: du couplage faible au couplage fort

The first chapters introduce briefly conformal theories, Moore and Seiberg polynomial equations and Gervais-Neveu quantization of Liouville theory. The next chapters present the original results of this thesis. First, the algebra of the chiral components is completely elucidated, both in the Bloch wave and in the Quantum Group basis. Then, in the strong coupling regime, the proof of the truncation theorem of Gervais to real weight operators for C = 7, 13, 19 is completed, especially including fractionnal spins. In this strong coupling regime, the new cosmological term yields a real string susceptibility equal to the real part of the KPZ formula. And eventually the N-point functions of a strongly coupled topological model are obtained.

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arXiv1995

Exchange statistics in 1D: from the viewpoint of exact solution

We show that the exchange statistics have consequences in 1D systems with compact topology, contrary to the common opinion that exchange statistics is arbitrary in 1D. As examples of non-trivial statistical behavior we exactly calculate the partition function and correlators for systems of free q-particles on compactified chains using functional integral techniques and the supersymmetric trick. In particular we consider a spin 1/2 XY-chain with periodic boundary conditions that corresponds to the case of q=-1.

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arXiv1997

KdV type systems and W-algebras in the Drinfeld-Sokolov approach

The generalized Drinfeld-Sokolov construction of KdV systems is reviewed in the case of an arbitrary affine Lie algebra paying particular attention to Hamiltonian aspects and $\W$-algebras. Some extensions of known results as well as a new interpretation of the construction are also presented.

hep-th↗
arXiv1995

Is String Theory in Knots?

It is sometimes said that there may be a unique algebraic theory independent of space-time topologies which underlies superstring and p-brane theories. In this paper, I construct some algebras using knot relations within the framework of event-symmetric string theory, and ask the question "Is string theory in knots?"

hep-th↗
arXiv1995

Generalized Deformed Oscillators and Algebras

The generalized deformed oscillator schemes introduced as unified frameworks of various deformed oscillators are proved to be equivalent, their unified representation leading to a correspondence between the deformed oscillator and the N=2 supersymmetric quantum mechanics (SUSY-QM) scheme. In addition, several physical systems (two identical particles in two dimensions, isotropic oscillator and Kepler system in a 2-dim curved space) and mathematical structures (quadratic algebra QH(3), finite W algebra $\bar {\rm W}_0$) are shown to possess the structure of a generalized deformed su(2) algebra, the representation theory of which is known. Furthermore, the generalized deformed parafermionic oscillator is identified with the algebra of several physical systems (isotropic oscillator and Kepler system in 2-dim curved space, Fokas--Lagerstrom, Smorodinsky--Winternitz and Holt potentials) and mathematical constructions (generalized deformed su(2) algebra, finite W algebras $\bar {\rm W}_0$ and W$_3^{(2)}$). The fact that the Holt potential is characterized by the W$_3^{(2)}$ symmetry is obtained as a by-product.

hep-th↗
arXiv1996

Analog of Lie Algebra and Lie Group for Quantum Non-Hamiltonian Systems

Quantum mechanics of Hamiltonian (non-dissipative) systems uses Lie algebra and analytic group (Lie group). In order to describe non-Hamiltonian (dissipative) systems in quantum theory we need to use non-Lie algebra and analytic quasigroup (loop). The author derives that analog of Lie algebra for quantum non-Hamiltonian systems is commutant Lie algebra and analog of Lie group for these systems is analytic commutant associative loop (Valya loop). A commutant Lie algebra is an algebra such that commutant (a subspace which is generated by all commutators) is a Lie subalgebra. Valya loop is a non-associative loop such that the commutant of this loop is associative subloop (group). We prove that a tangent algebra of Valya loop is a commutant Lie algebra. It is shown that generalized Heisenberg-Weyl algebra, suggested by the author to describe quantum non-Hamiltonian (dissipative) systems, is a commutant Lie algebra. As the other example of commutant Lie algebra, it is considered a generalized Poisson algebra for differential 1-forms. Note that non-Hamiltonian (dissipative) quantum theory has a broad range of application for non-critical strings in "coupling constant" phase space and bosonic string in non-Riemannian (for example, affine-metric) curved space which are non-Hamiltonian (dissipative) systems.

hep-th↗
arXiv1996

On the enumeration of irreducible k-fold Euler sums and their roles in knot theory and field theory

A generating function is given for the number, $E(l,k)$, of irreducible $k$-fold Euler sums, with all possible alternations of sign, and exponents summing to $l$. Its form is remarkably simple: $\sum_n E(k+2n,k) x^n = \sum_{d|k}μ(d) (1-x^d)^{-k/d}/k$, where $μ$ is the Möbius function. Equivalently, the size of the search space in which $k$-fold Euler sums of level $l$ are reducible to rational linear combinations of irreducible basis terms is $S(l,k) = \sum_{n<k}{\lfloor(l+n-1)/2\rfloor\choose n}$. Analytical methods, using Tony Hearn's REDUCE, achieve this reduction for the 3698 convergent double Euler sums with $l\leq44$; numerical methods, using David Bailey's MPPSLQ, achieve it for the 1457 convergent $k$-fold sums with $l\leq7$; combined methods yield bases for all remaining search spaces with $S(l,k)\leq34$. These findings confirm expectations based on Dirk Kreimer's connection of knot theory with quantum field theory. The occurrence in perturbative quantum electrodynamics of all 12 irreducible Euler sums with $l\leq 7$ is demonstrated. It is suggested that no further transcendental occurs in the four-loop contributions to the electron's magnetic moment. Irreducible Euler sums are found to occur in explicit analytical results, for counterterms with up to 13 loops, yielding transcendental knot-numbers, up to 23 crossings.

hep-th↗
arXiv1996

Parasupersymmetric Quantum Mechanics with Generalized Deformed Parafermions

A superposition of bosons and generalized deformed parafermions corresponding to an arbitrary paraquantization order $p$ is considered to provide deformations of parasupersymmetric quantum mechanics. New families of parasupersymmetric Hamiltonians are constructed in connection with two examples of su(2) nonlinear deformations such as introduced by Polychronakos and Ro\v cek.

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arXiv1996

Symmetry, Integrable Chain Models and Stochastic Processes

A general way to construct chain models with certain Lie algebraic or quantum Lie algebraic symmetries is presented. These symmetric models give rise to series of integrable systems. As an example the chain models with $A_n$ symmetry and the related Temperley-Lieb algebraic structures and representations are discussed. It is shown that corresponding to these $A_n$ symmetric integrable chain models there are exactly solvable stationary discrete-time (resp. continuous-time) Markov chains whose spectra of the transition matrices (resp. intensity matrices) are the same as the ones of the corresponding integrable models.

hep-th↗
arXiv1996

Statistics and Quantum Group Symmetries

Using twisted realizations of the symmetric groups, we show that Bose and Fermi statistics are compatible with transformations generated by compact quantum groups of Drinfel'd type.

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arXiv1996

A q-Deformation of the Harmonic Oscillator

The q-deformed harmonic oscillator is studied in the light of q-deformed phase space variables. This allows a formulation of the corresponding Hamiltonian in terms of the ordinary canonical variables $x$ and $p$. The spectrum shows unexpected features such as degeneracy and an additional part that cannot be reached from the ground state by creation operators. The eigenfunctions show lattice structure, as expected.

hep-th↗
arXiv1996

Universal R-matrix and Quantum Volterra Model

In this paper we explicitly prove that Integrable System solved by Quantum Inverse Scattering Method can be described with the pure algebraic object (Universal R-matrix) and proper algebraic representations. Namely, on the example of the Quantum Volterra model we construct L-operator and fundamental R--matrix from universal R--matrix for Quantum Affine $U_q(\widehat{sl}_2)$ Algebra and q-oscillator representation for it. In this way there exists an equivalence between the Integrable System with symmetry algebra A and the representation of this algebra.

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