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

Anna Beliakova; Marco De Renzi; Quentin Faes

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arXiv · 2512.15395

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

Anna Beliakova·Marco De Renzi·Quentin Faes

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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.

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Anna Beliakova, Marco De Renzi, Quentin Faes. 2025-12-17. Quantum Invariants of Ribbon Surfaces in $4$-Dimensional $2$-Handlebodies. https://arxiv.org/abs/2512.15395

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