Hierarchies of invariant spin models

TL;DR

This paper introduces hierarchical classes of state sum models based on SU(2) recoupling theory, unifying known invariants across dimensions and establishing their independence from triangulation choices, thus providing a broad framework for topological invariants.

Contribution

It develops a hierarchical framework of invariant spin models using recoupling theory, extending known models and proving their topological invariance across dimensions.

Findings

Unified hierarchy of spin models across dimensions

All models are triangulation-independent

Extension to PL-pairs with boundary triangulations

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.