1,327 research outputs found
Generating series and asymptotics of classical spin networks
We study classical spin networks with group SU(2). In the first part, using
gaussian integrals, we compute their generating series in the case where the
networks are equipped with holonomies; this generalizes Westbury's formula. In
the second part, we use an integral formula for the square of the spin network
and perform stationary phase approximation under some non-degeneracy
hypothesis. This gives a precise asymptotic behavior when the labels are
rescaled by a constant going to infinity.Comment: 33 pages, 3 figures; in version 2 added one reference and a comment
on the hypotheses of Theorem 1.
On SL(2, C) quantum 6j-symbol and its relation to the hyperbolic volume
We generalize the colored Alexander invariant of knots to an invariant of
graphs, and we construct a face model for this invariant by using the
corresponding 6j-symbol, which comes from the non-integral representations of
the quantum group U_q(sl_2). We call it the SL(2, C) quantum 6j-symbol, and
show its relation to the hyperbolic volume of a truncated tetrahedron.Comment: 30 pages, typos, argument for the R-matrix is modified, Sections 2
and 3 are exchange
Motion among random obstacles on a hyperbolic space
We consider the motion of a particle along the geodesic lines of the
Poincar\`e half-plane. The particle is specularly reflected when it hits
randomly-distributed obstacles that are assumed to be motionless. This is the
hyperbolic version of the well-known Lorentz Process studied by Gallavotti in
the Euclidean context. We analyse the limit in which the density of the
obstacles increases to infinity and the size of each obstacle vanishes: under a
suitable scaling, we prove that our process converges to a Markovian process,
namely a random flight on the hyperbolic manifold.Comment: 19 pages, 4 figure
6j-symbols, hyperbolic structures and the Volume Conjecture
We compute the asymptotical growth rate of a large family of
-symbols and we interpret our results in geometric terms by relating them
to volumes of hyperbolic truncated tetrahedra. We address a question which is
strictly related with S.Gukov's generalized volume conjecture and deals with
the case of hyperbolic links in connected sums of . We answer
this question for the infinite family of fundamental shadow links.Comment: 17 pages, 3 figures. Published on Geometry & Topology 11 (2007
Some remarks on the unrolled quantum group of sl(2)
In this paper we consider the representation theory of a non-standard
quantization of sl(2). This paper contains several results which have
applications in quantum topology, including the classification of projective
indecomposable modules and a description of morphisms between them. In the
process of proving these results the paper acts as a survey of the known
representation theory associated to this non-standard quantization of sl(2).
The results of this paper are used extensively in [arXiv:1404.7289] to study
Topological Quantum Field Theory (TQFT) and have connections with Conformal
Field Theory (CFT).Comment: 25 pages, v3: several mistakes corrected in the formulas of modified
trace
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