[Paper Review] Fusion 2-categories and a state-sum invariant for 4-manifolds

Christopher L. Douglas, David Reutter|arXiv (Cornell University)|Dec 31, 2018

Algebraic structures and combinatorial models48 references102 citations

TL;DR

The paper develops fusion 2-categories and spherical prefusion 2-categories, and constructs a state-sum invariant for oriented singular 4-manifolds that generalizes several known 4-manifold invariants.

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.

Motivation & Objective

• Motivate the search for a unified higher-categorical framework for 4D topological invariants and fully dualizable objects in higher categories.
• Introduce semisimple and fusion 2-categories and establish their relationship to multifusion categories as module 2-categories.
• Define sphericality for fusion 2-categories and show how it yields a 4-manifold state-sum invariant.
• Present a general state-sum formula and show specialization to known 4D invariants.
• Demonstrate invariance of the state sum under triangulation changes via bistellar moves.

Proposed method

• Define semisimple and presemisimple 2-categories and develop idempotent completion theory for 2-categories.
• Introduce fusion 2-categories as finite semisimple monoidal 2-categories with duals and simple unit.
• Develop sphericality via 2-spherical traces in pivotal 2-categories and establish spherical fusion 2-categories.
• Construct a state-sum Z_C(K) for oriented singular combinatorial 4-manifolds using simple objects and simple 1-morphisms labeled on simplices, with a 10j-type symbol Z(Γ).
• Prove invariance under labeling changes, vertex reordering, and bistellar moves to ensure a piecewise-linear invariant.

Experimental results

Research questions

• RQ1 How can fusion 2-categories be defined to yield a fully dualizable structure suitable for 4D invariants?
• RQ2 What conditions (sphericality) on fusion 2-categories allow construction of a state-sum invariant of singular 4-manifolds?
• RQ3 How does the resulting 4D state-sum relate to and unify existing invariants such as Crane–Yetter–Kauffman, Yetter–Dijkgraaf–Witten, Mackaay, and Cui?
• RQ4 What is the precise combinatorial framework (labels, dimensions, and 10j symbols) needed for the state-sum?
• RQ5 Is the state-sum invariant independent of triangulation and combinatorial structure?

Key findings

• Introduces semisimple 2-categories and proves they correspond to modules over multifusion categories.
• Defines fusion 2-categories and shows they generalize 1-categorical fusion categories to 2-categorical settings.
• Formulates sphericality for fusion 2-categories using 2-spherical traces and demonstrates it yields a well-defined 4D state-sum invariant.
• Constructs the state-sum Z_C(K) for oriented singular combinatorial 4-manifolds using simple object and simple 1-morphism labels and a 10j symbol derived from the fusion 2-category.
• Shows that the state-sum specializes to known 4D invariants in appropriate cases (Crane–Yetter–Kauffman, twisted Yetter–Dijkgraaf–Witten, Mackaay, Cui).
• Proves the invariant is independent of labeling skeleton, vertex ordering, and bistellar moves, yielding a piecewise-linear invariant of singular 4-manifolds.