Invariants of Spin Networks with Boundary in Quantum Gravity and TQFTs

Abstract

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

Hierarchies of invariant spin models

“…In this case the labelling j have to be assigned to the (d − 2)simplices of each triangulation (namely edges in d = 3, triangles in d = 4, and so on). Thus we recover the Ponzano-Regge model Z 3 P R and the Ooguri-Crane-Yetter invariant Z 4 CY (q = 1); the other invariants are new. A similar remark holds true also in the q-deformed context, where the hierarchy would be rewritten in terms of the counterparts Z d (q) ≡ Z[M d ](q) (and in particular we found the Turaev-Viro invariant in d = 4, see [18]).…”

“…• step 2) The state sum for (T d , ∂T d ) gives rise to a P L-invariant Z[(M d , ∂M d )] owing to the fact that we can exploit a set of topological moves, the elementary shellings of Pachner [10] (the algebraic identities associated with such moves in d = 3 were established in [2] and [3]). …”

“…Thus, as an illustration of the analytical approach, we present in Section 2 a short rewiew of [2] and [3], while in Section 3 we provide the extension of the CY -invariant (q = 1) to the case of a pair (M 4 , ∂M 4 ) and the expression of the induced Z 3 χ on the boundary. In the following Section 4 we give the explicit form of…”

“…and induced 2-dimensional invariant Following [2] and [3], the connection between a recoupling scheme of SU(2) angular momenta and the combinatorial structure of a compact, 3-dimensional simplicial pair (M 3 , ∂M 3 ) can be established by considering colored triangulations which allow us to specialize the map (2) according to…”

Implementing holographic projections in Ponzano–Regge gravity

“…Since we can freely choose the decoupling parameter R ≫ J(≤ L) at this level, we carry out the rescaling both on the three types of 6j symbols according to (20), (22), (24) and on the N * 1 phase factors and weights (2J +1) ∼ (2R) in (15) and (18). Then the functional which turns out to be associated with the resulting configuration is…”