Invariants of Spin Networks with Boundary in Quantum Gravity and TQFTs

Abstract

The search for classical or quantum combinatorial invariants of compact n-dimensional manifolds (n = 3, 4) plays a key role both in topological field theories and in lattice quantum gravity (see, e.g., [1–4]). 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 3 jm 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 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 for 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 3 jm symbols, each of them being associated with a triangle in the surface.