63 research outputs found
Link Invariants and Combinatorial Quantization of Hamiltonian Chern-Simons Theory
We define and study the properties of observables associated to any link in
(where is a compact surface) using the
combinatorial quantization of hamiltonian Chern-Simons theory. These
observables are traces of holonomies in a non commutative Yang-Mills theory
where the gauge symmetry is ensured by a quantum group. We show that these
observables are link invariants taking values in a non commutative algebra, the
so called Moduli Algebra. When these link invariants are pure
numbers and are equal to Reshetikhin-Turaev link invariants.Comment: 39, latex, 7 figure
Combinatorial expression for universal Vassiliev link invariant
The most general R-matrix type state sum model for link invariants is
constructed. It contains in itself all R-matrix invariants and is a generating
function for "universal" Vassiliev link invariants. This expression is more
simple than Kontsevich's expression for the same quantity, because it is
defined combinatorially and does not contain any integrals, except for an
expression for "the universal Drinfeld's associator".Comment: 20 page
More on quantum groups from the the quantization point of view
Star products on the classical double group of a simple Lie group and on
corresponding symplectic grupoids are given so that the quantum double and the
"quantized tangent bundle" are obtained in the deformation description.
"Complex" quantum groups and bicovariant quantum Lie algebras are discused from
this point of view. Further we discuss the quantization of the Poisson
structure on symmetric algebra leading to the quantized enveloping
algebra as an example of biquantization in the sense of Turaev.
Description of in terms of the generators of the bicovariant
differential calculus on is very convenient for this purpose. Finally
we interpret in the deformation framework some well known properties of compact
quantum groups as simple consequences of corresponding properties of classical
compact Lie groups. An analogue of the classical Kirillov's universal character
formula is given for the unitary irreducible representation in the compact
case.Comment: 18 page
Topological Quantum Field Theories and Operator Algebras
We review "quantum" invariants of closed oriented 3-dimensional manifolds
arising from operator algebras.Comment: For proceedings of "International Workshop on Quantum Field Theory
and Noncommutative Geometry", Sendai, November 200
Mapping Class Group Actions on Quantum Doubles
We study representations of the mapping class group of the punctured torus on
the double of a finite dimensional possibly non-semisimple Hopf algebra that
arise in the construction of universal, extended topological field theories. We
discuss how for doubles the degeneracy problem of TQFT's is circumvented. We
find compact formulae for the -matrices using the canonical,
non degenerate forms of Hopf algebras and the bicrossed structure of doubles
rather than monodromy matrices. A rigorous proof of the modular relations and
the computation of the projective phases is supplied using Radford's relations
between the canonical forms and the moduli of integrals. We analyze the
projective -action on the center of for an
-st root of unity. It appears that the -dimensional
representation decomposes into an -dimensional finite representation and a
-dimensional, irreducible representation. The latter is the tensor product
of the two dimensional, standard representation of and the finite,
-dimensional representation, obtained from the truncated TQFT of the
semisimplified representation category of .Comment: 45 page
Some computations in the cyclic permutations of completely rational nets
In this paper we calculate certain chiral quantities from the cyclic
permutation orbifold of a general completely rational net. We determine the
fusion of a fundamental soliton, and by suitably modified arguments of A. Coste
, T. Gannon and especially P. Bantay to our setting we are able to prove a
number of arithmetic properties including congruence subgroup properties for
matrices of a completely rational net defined by K.-H. Rehren .Comment: 30 Pages Late
Extension of geodesic algebras to continuous genus
Using the Penner--Fock parameterization for Teichmuller spaces of Riemann
surfaces with holes, we construct the string-like free-field representation of
the Poisson and quantum algebras of geodesic functions in the continuous-genus
limit. The mapping class group acts naturally in the obtained representation.Comment: 16 pages, submitted to Lett.Math.Phy
Galois currents and the projective kernel in Rational Conformal Field Theory
The notion of Galois currents in Rational Conformal Field Theory is
introduced and illustrated on simple examples. This leads to a natural
partition of all theories into two classes, depending on the existence of a
non-trivial Galois current. As an application, the projective kernel of a RCFT,
i.e. the set of all modular transformations represented by scalar multiples of
the identity, is described in terms of a small set of easily computable
invariants
On intermediate subfactors of Goodman-de la Harpe-Jones subfactors
In this paper we present a conjecture on intermediate subfactors which is a
generalization of Wall's conjecture from the theory of finite groups. Motivated
by this conjecture, we determine all intermediate subfactors of
Goodman-Harpe-Jones subfactors, and as a result we verify that
Goodman-Harpe-Jones subfactors verify our conjecture. Our result also gives a
negative answer to a question motivated by a conjecture of
Aschbacher-Guralnick.Comment: To appear in Comm. Math. Phy
Quantum geometry from 2+1 AdS quantum gravity on the torus
Wilson observables for 2+1 quantum gravity with negative cosmological
constant, when the spatial manifold is a torus, exhibit several novel features:
signed area phases relate the observables assigned to homotopic loops, and
their commutators describe loop intersections, with properties that are not yet
fully understood. We describe progress in our study of this bracket, which can
be interpreted as a q-deformed Goldman bracket, and provide a geometrical
interpretation in terms of a quantum version of Pick's formula for the area of
a polygon with integer vertices.Comment: 19 pages, 11 figures, revised with more explanations, improved
figures and extra figures. To appear GER
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