Recent Developments in General Relativity pp 419-426 | Cite as

# 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, 2, 3, 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 6*j* 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.

## Keywords

Spin Variable Spin Network Topological Field Theory Inverse Move Empty Boundary## Preview

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