Hierarchies of invariant spin models

TL;DR

The paper develops a dimension-spanning framework of PL-invariant spin models built from SU(2) recoupling theory, unifying known invariants such as Ponzano–Regge in 3D, Crane–Yetter in 4D, and Turaev–Viro under a common hierarchical construction. By labeling simplices with higher j-symbols (specifically the $igl\{3(d-2)(d+1)/2\bigr\}j$ blocks) and proving invariance under Pachner moves and boundary shellings, the authors derive invariant state sums for closed manifolds and for PL-pairs $(M^d,\partial M^d)$ in arbitrary dimension. They present two principal routes to closed $M^d$ invariants: colorings of $(d-1)$-simplices yielding $Z[M^d](q)=w_q^{2[1+(-1)^{d-1}\chi(M^d)]}$ and colorings of $(d-2)$-simplices leading to the same Euler-characteristic-driven structure, along with corresponding $q$-deformations. The work lays out an algorithmic, diagrammatic approach to generate PL-invariants across dimensions, with potential connections to BF theories and discretized quantum gravity in higher dimensions.

Abstract

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 $(T^d,\partial T^d)$ of a PL-pair $(M^d,\partial M^d)$, 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 PL-invariant models can be defined in any dimension d.

Figures (23)

• Figure 1: The shelling $[2 \rightarrow 2]^{{\bf 3}}_{sh}$ and the corresponding bistellar $[2 \rightarrow 2]^{{\bf 2}}_{bst}$.
• Figure 2: The shelling $[1 \rightarrow 3]^{{\bf 3}}_{sh}$ and the corresponding bistellar $[1 \rightarrow 3]^{{\bf 2}}_{bst}$.
• Figure 3: The shelling $[3 \rightarrow 1]^{{\bf 3}}_{sh}$ and the corresponding bistellar $[3 \rightarrow 1]^{{\bf 2}}_{bst}$.
• Figure 4: Diagrammatic representation of the 15j symbol of the second type.
• Figure 5: Diagrammatic representation of the move $[1\rightarrow 4]_{sh}^{{\bf 4}}$.