Advances in Theoretical and Mathematical Physics

Volume 18 (2014)

Number 3

Theory of intersecting loops on a torus

Pages: 709 – 740



J. E. Nelson (Dipartimento di Fisica, Sezione Teorica, Università degli Studi di Torino, Italy; and INFN, Sezione di Torino, Italy)

R. F. Picken (Departamento de Matem´atica and CAMGSD, Centro de Análise Matemática, Geometria e Sistemas Dinâmicos, Instituto Superior Técnico, Universidade de Lisboa, Portugal)


We continue our investigation into intersections of closed paths on a torus, to further our understanding of the commutator algebra of Wilson loop observables in $2 + 1$ quantum gravity, when the cosmological constant is negative.We give a concise review of previous results, e.g. that signed area phases relate observables assigned to homotopic loops, and present new developments in this theory of intersecting loops on a torus. We state precise rules to be applied at intersections of both straight and crooked/rerouted paths in the covering space $\mathbb{R}^2$. Two concrete examples of combinations of different rules are presented.

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