• Corpus ID: 2132340

6j symbols for U_q(sl_2) and non-Euclidean tetrahedra

  title={6j symbols for U\_q(sl\_2) and non-Euclidean tetrahedra},
  author={Yuka U. Taylor and Chris T. Woodward},
  journal={arXiv: Quantum Algebra},
We relate the semiclassical asymptotics of the 6j symbols for the representation theory of the quantized enveloping algebra U_q(sl_2) at q a primitive root of unity, or q positive real, to the geometry of non-Euclidean tetrahedra. The formulas are motivated by the geometry of conformal blocks in the Wess-Zumino-Witten model; they generalize formulas in the case q = 1 of Wigner, Ponzano and Regge, and Schulten and Gordon, proved by J. Roberts. 

Figures from this paper

6j–symbols, hyperbolic structures and the volume conjecture
We compute the asymptotical growth rate of a large family of $U_q(sl_2)$ $6j$-symbols and we interpret our results in geometric terms by relating them to volumes of hyperbolic truncated tetrahedra.
6j –symbols, hyperbolic structures and the volume
We compute the asymptotical growth rate of a large family of Uq.sl2/ 6j –symbols and we interpret our results in geometric terms by relating them to volumes of hyperbolic truncated tetrahedra. We
Recurrence relation for the 6j-symbol of suq(2) as a symmetric eigenvalue problem
A well-known recurrence relation for the 6j-symbol of the quantum group suq(2) is realized as a tridiagonal, symmetric eigenvalue problem. This formulation can be used to implement an efficient
Closure constraints for hyperbolic tetrahedra
We investigate the generalization of loop gravity's twisted geometries to a q-deformed gauge group. In the standard undeformed case, loop gravity is a formulation of general relativity as a
On the quantization of polygon spaces
Moduli spaces of polygons have been studied since the nineties for their topological and symplectic properties. Under generic assumptions, these are symplectic manifolds with natural global
Encoding Curved Tetrahedra in Face Holonomies: Phase Space of Shapes from Group-Valued Moment Maps
We present a generalization of Minkowski’s classic theorem on the reconstruction of tetrahedra from algebraic data to homogeneously curved spaces. Euclidean notions such as the normal vector to a
Toeplitz operators in TQFT via skein theory
Topological quantum field theory associates to a punctured surface Σ, a level r and colors c in {1, . . . , r− 1} at the marked points a finite dimensional hermitian space Vr(Σ, c). Curves γ on Σ act
Quantum edge modes in 3d gravity and 2+1d topological phases of matter
We analyze the edge mode structure of Euclidean three dimensional gravity from within the quantum theory as embodied by a Ponzano-Regge-Turaev-Viro discrete state sum with Gibbons-=-Hawking-York
The Regge symmetry, confocal conics, and the Schläfli formula
The Regge symmetry is a set of remarkable relations between two tetrahedra whose edge lengths are related in a simple fashion. It was first discovered as a consequence of an asymptotic formula in


Classical 6j-symbols and the tetrahedron
A classical 6j-symbol is a real number which can be associated to a labelling of the six edges of a tetrahedron by irreducible representations of SU(2). This abstract association is traditionally
Asymptotics and 6j-symbols
Recent interest in the Kashaev-Murakami-Murakami hyperbolic volume conjecture has made it seem important to be able to understand the asymptotic behaviour of certain special functions arising from
3-Dimensional Gravity and the Turaev-Viro Invariant
We derived an asymptotic formula for q-6j symbol. This is a generalization of the former work by Ponzano and Regge. Studying the q-deformed su(2) spin network as a 3-dimensional quantum gravity
Computer calculation of Witten's 3-manifold invariant
Witten's 2+1 dimensional Chern-Simons theory is exactly solvable. We compute the partition function, a topological invariant of 3-manifolds, on generalized Seifert spaces. Thus we test the path
Eigenvalues of products of unitary matrices and quantum Schubert calculus
We describe the inequalities on the possible eigenvalues of products of unitary matrices in terms of quantum Schubert calculus. Related problems are the existence of flat connections on the punctured
State sum invariants of 3 manifolds and quantum 6j symbols
A volume formula for hyperbolic tetrahedra in terms of edge lengths
Abstract.We give a closed formula for volumes of generic hyperbolic tetrahedra in terms of edge lengths. The cue of our formula is by the volume conjecture for the Turaev-Viro invariant of closed
Semiclassical approximations to 3j- and 6j-coefficients for quantum­ mechanical coupling of angular momenta An inverse problem in statistical mechanics Direct determination of the Iwasawa decomposition for noncompact
The coupling of angular momenta is studied using quantum mechanics in the limit of large quantum numbers (semiclassical limit). Uniformly valid semiclassical expressions are derived for the 3j
Geometric quantization of Chern-Simons gauge theory
We present a new construction of the quantum Hubert space of ChernSimons gauge theory using methods which are natural from the threedimensional point of view. To show that the quantum Hubert space
Quantum generalization of the Horn conjecture
(Recall that the Lie algebra of the special unitary group SU(n) is isomorphic to the real vector space of traceless Hermitian matrices as representations of SU(n) and hence the terminology "additive