[Paper Review] Uniqueness of unitary structure for unitarizable fusion categories

David Reutter|arXiv (Cornell University)|Jun 24, 2019

Algebraic structures and combinatorial models4 citations

TL;DR

This paper proves that every unitarizable fusion category—along with semisimple C*-tensor categories, braided fusion categories, and module categories—admits a unique unitary structure up to unitary monoidal equivalence. The proof relies on a categorified polar decomposition theorem, showing that any monoidal equivalence decomposes uniquely into a unitary equivalence followed by a positive auto-equivalence, which must be trivial due to finiteness constraints, thereby establishing uniqueness of the unitary structure.

ABSTRACT

We show that every unitarizable fusion category, and more generally every semisimple C*-tensor category, admits a unique unitary structure. Our proof is based on a categorified polar decomposition theorem for monoidal equivalences between such categories. We prove analogous results for unitarizable braided fusion categories and module categories.

Motivation & Objective

• To resolve the open problem of whether unitarizable fusion categories admit a unique unitary structure.
• To extend uniqueness results to semisimple C*-tensor categories, braided fusion categories, and module categories.
• To establish that monoidal equivalences between unitary fusion categories are uniquely unitary up to isomorphism.
• To show that natural isomorphisms between unitary monoidal functors are also uniquely unitary.
• To demonstrate that the forgetful 2-functor from unitary to general monoidal categories is an equivalence, confirming unitarizability as a property, not structure.

Proposed method

• Categorified polar decomposition: every monoidal equivalence between semisimple C*-categories factors uniquely into a unitary monoidal equivalence followed by a positive monoidal auto-equivalence.
• Leveraging finiteness of the group of monoidal auto-equivalences in fusion categories to show that positive auto-equivalences are trivial.
• Extending the polar decomposition to natural isomorphisms between functors, decomposing them into unitary and positive parts.
• Using the finiteness of the universal grading group to rule out non-trivial positive monoidal natural automorphisms.
• Adapting the framework to braided categories by showing the constructions are compatible with braidings.
• Translating the method to module categories by replacing monoidal auto-equivalences with module auto-equivalences, preserving the uniqueness argument.

Experimental results

Research questions

• RQ1 Can a unitarizable fusion category admit more than one unitary structure up to monoidal equivalence?
• RQ2 Is the unitary structure on a semisimple C*-tensor category uniquely determined?
• RQ3 Are monoidal natural isomorphisms between unitary monoidal functors necessarily unitary?
• RQ4 Does a braided unitarizable fusion category admit a unique unitary braided structure?
• RQ5 Is the forgetful 2-functor from unitary to general monoidal categories an equivalence?

Key findings

• Every monoidal equivalence between unitary fusion categories is monoidally naturally isomorphic to a unitary monoidal equivalence, proving uniqueness of unitary structure.
• Every monoidal natural isomorphism between unitary monoidal functors is unitary, due to the triviality of positive natural automorphisms.
• The 2-groupoid of unitarizable fusion categories is equivalent to the 2-groupoid of unitary fusion categories under unitary equivalences, confirming unitarizability as a property.
• The forgetful 2-functor from unitary to general monoidal categories is an equivalence, showing that unitary structures are unique up to isomorphism.
• The results extend to braided fusion categories: every braided monoidal equivalence is isomorphic to a unitary braided equivalence, ensuring unique unitary braided structures.
• The same uniqueness holds for module categories over unitary fusion categories, with module equivalences uniquely unitarizable.