Invariants of spin networks with boundary in Quantum Gravity and TQFT's

TL;DR

This paper generalizes the Ponzano-Regge state sum to 3-manifolds with boundary, creating invariants that incorporate boundary data and revealing a common algebraic structure across different models in quantum topology and gravity.

Contribution

It introduces a boundary-inclusive state sum model for 3D manifolds, extending previous invariants and establishing a framework for higher-dimensional generalizations.

Findings

The new state sum includes boundary data via Wigner symbols.

The model demonstrates topological invariance through algebraic identities.

Extensions to 2D and 4D models are feasible and outlined.

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. We present here a generalization of the partition function proposed by Ponzano and Regge to the case of a compact 3-dimensional simplicial pair $(M^3, \partial M^3)$. The resulting state sum $Z[(M^3, \partial M^3)]$ contains both Racah-Wigner 6j symbols associated with tetrahedra and Wigner 3jm symbols associated with triangular faces lying in $\partial M^3$. 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.