The central result is a formal classification. For each bordism bicategory covered by the classification—a formal setting for tracking geometric pieces and how they compose—all linear representations arise as Reshetikhin–Turaev theories. In this setting, a representation is a formal rule for assigning algebraic data to those geometric pieces. The paper also gives a direct extension of the construction to circles and proves that it is a symmetric monoidal functor in the stated decomposed-signature setting. In plain language, the assignments respect the prescribed ways of composing and combining the pieces.
A geometric construction with a concrete calculation route
The mathematical input is a modular tensor category—a formal algebraic structure used to encode the theory—together with a choice of square root of its global dimension. The construction combines internal string diagrams for objects and 1-morphisms with Reshetikhin–Turaev ribbon invariants for 2-morphisms. In effect, the paper uses diagrams for the object and 1-morphism levels and ribbon invariants for the 2-morphism level.
For the three-dimensional part, the images of 2-morphisms are described as computable from surgery presentations of 3-manifolds. A surgery presentation is the geometric recipe used to specify the manifold for the calculation. The paper’s claim is therefore concrete: the 2-morphism images can be obtained from the stated presentation.
Two formal checks on the construction
The construction is compared with two other formulations. On the relevant category of parametrized closed surfaces, it agrees with the original Reshetikhin–Turaev functor under the equivalence specified in the paper. On the half-signature setting, the constructed functor, after the presentation map is applied, is equivalent to the generators-and-relations functor ZC. That comparison connects the geometric construction with a formulation built from formal generators and the relations among them.
One possible ambiguity concerns the square root used in the setup. On the half-signature subbicategory, the two square-root choices agree and give a single symmetric monoidal functor. That agreement is limited to the half-signature setting described in the paper.
How the representations are classified
The classification is organized around finite direct sums of modular tensor categories, meaning that several such categories can appear together as one formal representation. In the half-signature case, the categories in a direct sum must have equal anomalies. Here, anomaly is the formal quantity the paper uses to state this compatibility condition. For oriented bordisms, those direct sums are restricted to modular tensor categories with anomaly 1.
The paper then tracks how the required root choices change in related bordism variants. In the t-times extension, the data are finite direct sums of modular tensor categories with equal anomalies together with a single tth root of the anomaly. In the signature and p1 cases, the added choice is respectively a single square root or a 6th root of the anomaly.
Componentwise versions impose conditions separately across the direct-sum pieces. Componentwise half-signature representations are classified by finite direct sums of modular tensor categories. In the componentwise signature case, a square root of the anomaly is required for each direct summand.
A result bounded by its assumptions
The conclusions are formal theorems under the paper’s stated assumptions. The input is a modular tensor category with the square-root choice described above, and the classification applies to the listed bordism bicategories. The supplied analysis does not establish the same conclusions beyond those inputs and settings.
The functoriality result is conditional on the stated definitions and modular tensor category assumptions. Its scope is the decomposed-signature bordism setting specified by the construction.
At publication, the document was an arXiv version 1 preprint dated 26 August 2026; no journal or DOI was listed in the supplied metadata. The acknowledgements thank André Henriques for proposing the project and for continued guidance and support; funding is not reported.
Paper data and sources
Original title: All once-extended 3D TQFTs are Reshetikhin--Turaev theories
Authors: Glen Lim
Journal/Repository: arXiv
Status: Preprint, not yet peer-reviewed
First online: 2026-08-26
DOI: Not available
Original paper · Full text