Three mathematical faces of SU(2)-spin networks

Spin networks are at the core of quantum gravity [1]. We have neither the space nor the competence to give an exhaustive list of the physical and philosophical interpretations of this notion (for these, see for example [2] and [3]). New (and old) approaches towards creating a finite quantum theory of general relativity would use combinatorial expressions in Feynman integrals, spin networks, spin foams and others combinatorial objects. The intention is to get out the standard “continuous” geometry. Our aim is to plug the mathematical community at large into these procedures as participants. For this, because of the different cultural backrounds, we would like to change tack: to relate discrete (combinatorial) objects to the standard “continuous” geometry. From the mathematical point of view, relations of this type give rise to identifications between the geometry of varieties and combinatorial objects, as exemplified by the relation between Lie algebras and root systems, or toric varieties and polytopes. The general mathematical mechanism of such “interpretations” could be called the analytic theory of non-Abelian theta functions, since they run completely parallel to the original classical theory of theta functions. Recall that this classical theory has three parts: