Asymptotics of classical spin networks

Details Asymptotics of classical spin networks Asymptotics of classical spin networks

2009-02-18

arxiv.org/abs/0902.3113v4

Mathematics

Geometric Topology

24 pages, 32 figures

Scientific paper

A spin network is a cubic ribbon graph labeled by representations of $\mathrm{SU}(2)$. Spin networks are important in various areas of Mathematics (3-dimensional Quantum Topology), Physics (Angular Momentum, Classical and Quantum Gravity) and Chemistry (Atomic Spectroscopy). The evaluation of a spin network is an integer number. The main results of our paper are: (a) an existence theorem for the asymptotics of evaluations of arbitrary spin networks (using the theory of $G$-functions), (b) a rationality property of the generating series of all evaluations with a fixed underlying graph (using the combinatorics of the chromatic evaluation of a spin network), (c) rigorous effective computations of our results for some $6j$-symbols using the Wilf-Zeilberger theory, and (d) a complete analysis of the regular Cube $12j$ spin network (including a non-rigorous guess of its Stokes constants), in the appendix.

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