Rigidity of non-negligible objects of moderate growth in braided categories

TL;DR

This paper establishes conditions under which objects in braided categories are rigid, simplifying proofs of rigidity in vertex operator algebras and linking modular functors to modular fusion categories, with implications for non-semisimple categories.

Contribution

It proves that non-negligible objects with certain dimension bounds are automatically rigid and connects modular functors to modular fusion categories, extending rigidity results to non-semisimple contexts.

Findings

Non-negligible objects with bounded endomorphism dimensions are rigid.

Semisimple categories of moderate growth are rigid if weakly rigid.

Quantum traces of nilpotent endomorphisms vanish in rigid moderate growth categories.

Abstract

Let $k$ be a field, and let $\mathcal{C}$ be a Cauchy complete $k$-linear braided category with finite dimensional morphism spaces and ${{\rm End}(\bf 1)}=k$. We call an indecomposable object $X$ of $\mathcal C$ non-negligible if there exists $Y\in \mathcal{C}$ such that $\bf 1$ is a direct summand of $Y\otimes X$. We prove that every non-negligible object $X\in \mathcal{C}$ such that ${\rm dim}{\rm End}(X^{\otimes n})<n!$ for some $n$ is automatically rigid. In particular, if $\mathcal{C}$ is semisimple of moderate growth and weakly rigid, then $\mathcal{C}$ is rigid. As applications, we simplify Huang's proof of rigidity of representation categories of certain vertex operator algebras, and we get that for a finite semisimple monoidal category $\mathcal{C}$, the data of a $\mathcal{C}$-modular functor is equivalent to a modular fusion category structure on $\mathcal{C}$, answering a question of Bakalov and Kirillov. Finally, we show that if $\mathcal{C}$ is rigid and has moderate growth, then the quantum trace of any nilpotent endomorphism in $\mathcal{C}$ is zero. Hence $\mathcal{C}$ admits a semisimplification, which is a semisimple braided tensor category of moderate growth. Finally, we discuss rigidity in braided r-categories which are not semisimple, which arise in logarithmic conformal field theory. These results allow us to simplify a number of arguments of Kazhdan and Lusztig.