Volume 17, issue 1 (2013)

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Asymptotics of classical spin networks

Stavros Garoufalidis and Roland van der Veen

Appendix: Don Zagier

Geometry & Topology 17 (2013) 1–37

A spin network is a cubic ribbon graph labeled by representations of 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 nonrigorous guess of its Stokes constants), in the appendix.

Spin networks, ribbon graphs, $6j$–symbols, Racah coefficients, angular momentum, asymptotics, G-functions, Kauffman bracket, Jones polynomial, Wilf-Zeilberger method, Borel transform, enumerative combinatorics, recoupling, Nilsson
Mathematical Subject Classification 2010
Primary: 57N10
Secondary: 57M25
Received: 20 February 2009
Revised: 9 May 2012
Accepted: 12 July 2012
Published: 5 February 2013
Proposed: Walter Neumann
Seconded: Vaughan Jones, Yasha Eliashberg
Stavros Garoufalidis
School of Mathematics
Georgia Institute of Technology
686 Cherry Street
Atlanta, GA 30332-0160
Roland van der Veen
Department of Mathematics
University of California
Berkeley, CA 94720-3840
Don Zagier
Max Planck Institute for Mathematics
Vivatsgasse 7, 53111