Quantum Invariants of Ribbon Surfaces in $4$-Dimensional $2$-Handlebodies

TL;DR

This paper develops a flexible quantum invariant framework for ribbon surfaces in 4-dimensional 2-handlebodies using unimodular ribbon categories, generalizing existing invariants and accommodating non-semisimple cases.

Contribution

It introduces a new approach to quantum invariants of ribbon surfaces in 4D handlebodies that does not require semisimplicity and extends the Reshetikhin-Turaev functor to labeled Kirby graphs.

Findings

Recoveries of known invariants by Bobtcheva-Messia, Broda-Petit, Gainutdinov-Geer-Patureau-Runkel, and Lee-Yetter.

A generalized functor applicable to non-semisimple categories and labeled Kirby graphs.

Invariants of framed links in boundary 3-manifolds up to 2-deformations.

Abstract

We use unimodular ribbon categories to construct quantum invariants of ribbon surfaces in $4$-dimensional $2$-handlebodies up to $1$-isotopy. In the process, we recover invariants due to Bobtcheva-Messia, Broda-Petit, Gainutdinov-Geer-Patureau-Runkel (in collaboration with the second author), and Lee-Yetter. Our approach does not assume semisimplicity, and is based on a generalization of the Reshetikhin-Turaev functor to the category of labeled Kirby graphs which also yields invariants of framed links in the boundary of $4$-dimensional $2$-handlebodies up to $2$-deformations. The setup is very flexible, and allows for several different constructions, using central elements satisfying equations introduced by Hennings and Bobtcheva-Messia, modified traces, and modules over Frobenius algebras satisfying conditions dictated by the diagrammatic calculus for embedded surfaces developed by Hughes, Kim, and Miller.