We introduce a quantum volume operator K in three-dimensional Quantum Gravity by taking into account a symmetrical coupling scheme of three SU(2) angular momenta. The spectrum of K is discrete and defines a complete set of eigenvectors which is alternative with respect to the complete sets employed when the usual binary coupling schemes of angular momenta are considered. Each of these states, that we call quantum bubbles, represents an interference of extended configurations which provides a rigorous meaning to the heuristic notion of quantum tetrahedron. We study the generalized recoupling coefficients connecting the symmetrical and the binary basis vectors, and provide an explicit recursive solution for such coefficients by analyzing also its asymptotic limit.

In this paper we generalize the partition function proposed by Ponzano and Regge in 1968 to the case of a compact 3-dimensional simplicial pair (M, ∂M). The resulting state sum Z[(M, ∂M)] contains both Wigner 6j symbols associated with tetrahedra and Wigner 3jm symbols associated with triangular faces lying in ∂M. In order to show the invariance of Z[(M, ∂M)] under PL-homeomorphisms we exploit some results due to Pachner on the equivalence of n-dimensional PL-pairs both under bistellar moves on n-simplices in the interior of M and under elementary boundary operations (shellings and inverse shellings) acting on n-simplices which have some component in ∂M. We find, in particular, the algebraic identities -involving a suitable number of Wigner symbols -which realize the complete set of Pachner's boundary operations in n = 3.The results established for the classical SU (2)-invariant Z [(M, ∂M)] are further extended to the case of the quantum enveloping algebra U q (sl(2, C)) (q a root of unity). The corresponding quantum invariant, Z q [(M, ∂M)], turns out to be the counterpart of the Turaev-Viro invariant for a closed 3-dimensional PL-manifold.

The search for classical or quantum combinatorial invariants of compact ndimensional manifolds (n = 3, 4) plays a key role both in topological field theories and in lattice quantum gravity (see e.g. [P-R], [T-V], [O92], [C-K-S]). We present here a generalization of the partition function proposed by Ponzano and Regge to the case of a compact 3-dimensional simplicial pair (M 3 , ∂M 3 ). The resulting state sum Z[(M 3 , ∂M 3 )] contains both Racah-Wigner 6j symbols associated with tetrahedra and Wigner 3jm symbols associated with triangular faces lying in ∂M 3 . The analysis of the algebraic identities associated with the combinatorial transformations involved in the proof of the topological invariance makes it manifest a common structure underlying the 3-dimensional models with empty and non empty boundaries respectively. The techniques developed in the 3-dimensional case can be further extended in order to deal with combinatorial models in n = 2, 4 and possibly to establish a hierarchy among such models. As an example we derive here a 2-dimensional closed state sum model including suitable sums of products of double 3jm symbols, each one of them being associated with a triangle in the surface.Recall that a closed P L-manifold of dimension n is a polyhedron M ∼ = |T |, each point of which has a neighborhood, in M, P L-homeomorphic to an open set in

In this paper we present classes of state sum models based on the recoupling theory of angular momenta of SU(2) (and of its q-counterpart U q (sl(2)), q a root of unity). Such classes are arranged in hierarchies depending on the dimension d, and include all known closed models, i.e. the Ponzano-Regge state sum and the Turaev-Viro invariant in dimension d = 3, the Crane-Yetter invariant in d = 4. In general, the recoupling coefficient associated with a d-simplex turns out to be a {3(d − 2)(d + 1)/2}j symbol, or its q-analog. Each of the state sums can be further extended to compact triangulations (, where the triangulation of the boundary manifold is not keeped fixed. In both cases we find out the algebraic identities which translate complete sets of topological moves, thus showing that all state sums are actually independent of the particular triangulation chosen. Then, owing to Pachner's theorems, it turns out that classes of P L-invariant models can be defined in any dimension d.

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