A universal state sum

TL;DR

This paper introduces a universal state sum framework that unifies many existing topological quantum field theory invariants by using specialized n-categories to construct (n+1)-dimensional TQFTs.

Contribution

It develops a general state sum construction that encompasses most known state sums, providing a unifying approach based on n-categories with specific properties.

Findings

Unifies various known state sum models under a single framework

Constructs (n+1)-dimensional TQFTs from n-categories

Provides a systematic method for deriving topological invariants

Abstract

We define a universal state sum construction which specializes to most previously known state sums (Turaev-Viro, Dijkgraaf-Witten, Crane-Yetter, Douglas-Reutter, Witten-Reshetikhin-Turaev surgery formula, Brown-Arf). The input data for the state sum is an n-category satisfying various conditions, including finiteness, semisimplicity and n-pivotality. From this n-category one constructs an n+1-dimensional TQFT, and applying the TQFT gluing rules to a handle decomposition of an n+1-manifold produces the state sum.