Fusion 2-categories and a state-sum invariant for 4-manifolds

TL;DR

The paper introduces semisimple $2$-categories, fusion $2$-categories, and spherical fusion $2$-categories, and constructs a state-sum invariant for oriented singular piecewise-linear $4$-manifolds from any spherical fusion $2$-category. It shows that finite semisimple $2$-categories are precisely the $2$-categories of finite semisimple module categories over multifusion categories, providing a natural route to monoidal structure via ordinary $2$-functors and ensuring that finite semisimple $2$-distributors are equivalent to $2$-functors. The work unifies earlier $4$-manifold invariants (e.g., Crane–Yetter, Yetter–Dijkgraaf–Witten, Cui) under a single higher-categorical framework and proves invariance properties such as the preservation of the normalized $10j$ action under state transposition, thereby enabling computable state-sum constructions. This yields a unified, computable approach to $4$-dimensional semisimple TFTs with potential applications in topological phases of matter and quantum computation.

Abstract

We introduce semisimple 2-categories, fusion 2-categories, and spherical fusion 2-categories. For each spherical fusion 2-category, we construct a state-sum invariant of oriented singular piecewise-linear 4-manifolds.

Theorems & Definitions (6)

• Definition 1
• Theorem 2
• Theorem 3
• Theorem 4
• Lemma I.{1}.1: State transposition preserves the normalized 10j action
• Lemma I.{1}.2: The copairing intertwines the associator state transposition